Vector 3d concepts 369 kho tài liệu bách khoa

KEY CONCEPTS DEFINITIONS: A VECTOR may be described as a quantity having both magnitude & direction A vector is generally represented by a directed line segment, say AB A is called the initial point & B is called the terminal point The magnitude of vector AB is expressed by AB ZERO VECTOR a vector of zero magnitude i.e.which has the same initial & terminal point, is called a ZERO VECTOR It is denoted by O UNIT VECTOR a vector of unit magnitude in direction of a vector a is called unit vector along a and is a denoted by aˆ symbolically aˆ a EQUAL VECTORS two vectors are said to be equal if they have the same magnitude, direction & represent the same physical quantity COLLINEAR VECTORS two vectors are said to be collinear if their directed line segments are parallel disregards to their direction Collinear vectors are also called PARALLEL VECTORS If they have the same direction they are named as like vectors otherwise unlike vectors Simbolically, two non zero vectors a and b are collinear if and only if, a K b , where K R COPLANAR VECTORS a given number of vectors are called coplanar if their line segments are all parallel to the same plane Note that “TWO VECTORS ARE ALWAYS COPLANAR” POSITION VECTOR let O be a fixed origin, then the position vector of a point P is the vector OP If a & b & position vectors of two point A and B, then , AB = b a = pv of B pv of A VECTOR ADDITION : If two vectors a & b are represented by OA & OB , then their sum a b is a vector represented by OC , where OC is the diagonal of the parallelogram OACB a (commutative) a b b a a a (a a b) c ( a) a ( b c) (associativity) ( a) a MULTIPLICATION OF VECTOR BY SCALARS : If a is a vector & m is a scalar, then m a is a vector parallel to a whose modulus is m times that of a This multiplication is called SCALAR MULTIPLICATION If a & b are vectors & m, n are scalars, then: m ( a ) ( a )m m a m ( n a ) n ( m a ) ( mn ) a (m n )a ma na m (a b) ma mb SECTION FORMULA : If a & b are the position vectors of two points A & B then the p.v of a point which divides AB in the ratio m : n is given by : r na m b a b Note p.v of mid point of AB = m n ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) DIRECTION COSINES : Let a a1ˆi a 2ˆj a 3kˆ the angles which this vector makes with the +ve directions OX,OY & OZ are called DIRECTION ANGLES & their cosines are called the DIRECTION COSINES a1 cos , a a2 cos , a3 cos a Note that, cos² + cos² + cos² =1 a VECTOR EQUATION OF A LINE : Parametric vector equation of a line passing through two point A (a ) & B( b) is given by, r a t (b a ) where t is a parameter If the line passes through the point A (a ) & is parallel to the vector b then its equation is, r a t b Note that the equations of the bisectors of the angles between the lines r = a + r = a +t b c b & r =a + c is : & r = a +p c b TEST OF COLLINEARITY : Three points A,B,C with position vectors a , b, c respectively are collinear, if & only if there exist scalars x , y, z not all zero simultaneously such that ; xa yb zc , where x + y + z = SCALAR PRODUCT OF TWO VECTORS : a.b a b cos (0 ), note that if is acute then a.b > a.a a & if is obtuse then a.b < a ,a.b b.a (commutative) (a a.b a b ˆi ˆi ˆj ˆj kˆ kˆ ; projection of a on b a.b a (b b 0) ˆi ˆj ˆj kˆ kˆ ˆi c) a b a c (distributive) b a b Note: That vector component of a along b = a b b and perpendicular to b = a – b b2 b a b the angle between a & b is given by cos if a a a1ˆi a 2ˆj a 3kˆ & a12 a 22 b1ˆi b ˆj b 3kˆ then a b = a1b1 + a2b2 + a3b3 b a 32 ab b , b 12 b22 b32 Note : (i) Maximum value of a b = (ii) Minimum values of a b = a b = a b a b ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) (iii) Any vector a can be written as , a = a i i (iv) A vector in the direction of the bisector of the angle between the two vectors a & b is a.j j bisector of the angle between the two vectors a & b is exterior angle between a & b is a b , a.k k a a a b b Hence R+ Bisector of the b , where R+ VECTOR PRODUCT OF TWO VECTORS : is the angle between them then a If a & b are two vectors & (i) b a b sin n , where n is the unit vector perpendicular to both a & b such that a , b & n forms a right handed screw system (ii) Lagranges Identity : for any two vectors a & b ;(a x b)2 (iii) Formulation of vector product in terms of scalar product: a b (a b ) a a a b a b b.b The vector product a x b is the vector c , such that (i) | c | = a 2b2 (a b) (ii) c a = 0; c b =0 and (iii) a , b, c form a right handed system (iv) a & b are parallel (collinear) (not commutative) b a a b a b (ma ) b a ( mb ) m (a b ) a ( b c ) (a b ) ( a c ) ˆi ˆi ˆj ˆj kˆ kˆ a1ˆi a 2ˆj a 3kˆ & (a ,b 0) i.e a K b , where K is a scalar where m is a scalar (distributive) ˆi ˆj kˆ , ˆj kˆ b b1ˆi b ˆj b3kˆ then a b ˆi , kˆ ˆi ˆj ˆi ˆj a1 a b1 b kˆ a3 b3 (v) If a (vi) Geometrically a b = area of the parallelogram whose two adjacent sides are represented by a & b (vii) Unit vector perpendicular to the plane of a & b is nˆ a b a b A vector of magnitude ‘r’ & perpendicular to the palne of a & b is b a b a b If is the angle between a & b then sin a b ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) (viii) Vector area If a , b & c are the pv’s of points A, B & C then the vector area of triangle ABC = axb bx c cx a The points A, B & C are collinear if a x b bx c Area of any quadrilateral whose diagonal vectors are d & d is given by 10 cx a d1 x d 2 SHORTEST DISTANCE BETWEEN TWO LINES : If two lines in space intersect at a point, then obviously the shortest distance between them is zero Lines which not intersect & are also not parallel are called SKEW LINES For Skew lines the direction of the shortest distance would be perpendicular to both the lines The magnitude of the shortest distance vector would be equal to that of the projection of AB along the direction of the line of shortest distance, LM is parallel to p x q = AB (p x q) pxq 11 LM Pr ojection of AB on LM = Pr ojection of AB on px q (b a ) (p xq ) pxq The two lines directed along p & q will intersect only if shortest distance = i.e (b a ).(p x q ) i.e b i.e b a pq a lies in the plane containing p & q If two lines are given by r1 a Kb & r2 a2 b x(a Kb i.e they are parallel then , d a 1) b SCALAR TRIPLE PRODUCT / BOX PRODUCT / MIXED PRODUCT : The scalar triple product of three vectors a , b & c is defined as : a x b c a b c sin cos where is the angle between a & b & is the angle between a b & c It is also defined as [ a b c ] , spelled as box product Scalar triple product geometrically represents the volume of the parallelopiped whose three couterminous edges are represented by a , b & c i e V [ a b c ] In a scalar triple product the position of dot & cross can be interchanged i.e a ( b x c ) ( a x b ) c OR [ a b c ] [ b c a ] [ c a b ] a ( b x c) a ( cx b) i e [ a b c ] [a c b] a1 a a If a a1ˆi a 2ˆj a 3kˆ ; b In general , if a then a b c a1 l b1ˆi b 2ˆj b3 kˆ & c c1ˆi c 2ˆj c 3kˆ then [a b c] a 2m a1 b1 a2 b2 a3 b3 c1 c2 c3 If a , b , c are coplanar a 3n ; b l mn [a b c] b1 l b2 m b3n & c c1 l b1 b b c1 c2 c3 c2 m c3 n ; where , m & n are non coplanar vectors ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Scalar product of three vectors, two of which are equal or parallel is i.e [ a b c ] 0, Note : If a , b , c are non coplanar then [ a b c ] for right handed system & for left handed system [a b c] [Ka b c] [i j k] = K[ a b c ] [(a b) c d ] [a c d] [ b c d] The volume of the tetrahedron OABC with O as origin & the pv’s of A, B and C being a , b & c respectively is given by V [a b c] The positon vector of the centroid of a tetrahedron if the pv’s of its angular vertices are a , b , c & d are given by [a b c d] Note that this is also the point of concurrency of the lines joining the vertices to the centroids of the opposite faces and is also called the centre of the tetrahedron In case the tetrahedron is regular it is equidistant from the vertices and the four faces of the tetrahedron Remember that : a b b c c a = *12 & a b b c c a =2 a bc VECTOR TRIPLE PRODUCT : Let a , b , c be any three vectors, then the expression a triple product GEOMETRICAL INTERPRETATION OF a Consider the expression a (b (b c ) is a vector & is called a vector c) c ) which itself is a vector, since it is a cross product of two vectors (b a & ( b x c ) Now a x ( b x c ) is a vector perpendicular to the plane containing a & ( b x c ) but b x c is a vector perpendicular to the plane b & c , therefore a x ( b x c ) is a vector lies in the plane of b & c and perpendicular to a Hence we can express a x ( b x c ) in terms of b & c i.e a x ( b x c ) = xb yc where x & y are scalars a x ( b x c ) = (a c) b (a x b ) x c 13 (a x b) x c = (a c) b (a b) c a x ( b x c) LINEAR COMBINATIONS / Linearly Independence and Dependence of Vectors : Given a finite set of vectors a , b , c , then the vector r x a combination of a , b , c , for any x, y, z (a) ( b c) a y b z c is called a linear R We have the following results : FUNDAMENTALTHEOREM IN PLANE : Let a ,b be non zero , non collinear vectors Then any vector r coplanar with a ,b can be expressed uniquely as a linear combination of a ,b i.e There exist some unique x,y (b) R such that x a yb r FUNDAMENTAL THEOREM IN SPACE : Let a , b ,c be non zero, non coplanar vectors in space Then any vector r , can be uniquily expressed as a linear combination of a , b ,c i.e There exist some unique x,y R such that x a yb zc r ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) If x , x , x n are n non zero vectors, & k1, k2, .kn are n scalars & if the linear combination (c) k 1x k 2x k n x n k1 ,k k n then we say that vectors x , x , x n are LINEARLY INDEPENDENT VECTORS If x , x , x n are not LINEARLY INDEPENDENT then they are said to be LINEARLY DEPENDENT (d) vectors i.e if k x k x x , x , x n knx n & if there exists at least one kr then are said to be LINEARLY DEPENDENT Note : If a = 3i + 2j + 5k then a is expressed as a LINEAR COMBINATION of vectors ˆi , ˆj, kˆ Also , a , ˆi, ˆj, kˆ form a linearly dependent set of vectors In general , every set of four vectors is a linearly dependent system ˆi , ˆj , kˆ are LINEARLY INDEPENDENT set of vectors For K1ˆi K 2ˆj K 3kˆ K1 = = K2 = K3 a is parallel to b i.e a x b Two vectors a & b are linearly dependent linear dependence of a & b Conversely if a x b then a & b are linearly independent If three vectors a , b , c are linearly dependent, then they are coplanar i.e [ a , b, c ] [ a , b, c ] 14 , conversely, if , then the vectors are linearly independent COPLANARITY OF VECTORS : Four points A, B, C, D with position vectors a , b , c , d respectively are coplanar if and only if there exist scalars x, y, z, w not all zero simultaneously such that x a+ y b + z c + w d = where, x + y + z + w = 15 RECIPROCAL SYSTEM OF VECTORS : If a , b , c & a ' ,b ' ,c ' are two sets of non coplanar vectors such that a a '= b b '= c c '= then the two systems are called Reciprocal System of vectors Note : 16 (a) a'= bx c abc ; b' cx a a bc ; c' axb abc EQUATION OF A PLANE : The equation ( r r0 ).n represents a plane containing the point with p.v r0 where n is a (b) vector normal to the plane r n d is the general equation of a plane Angle between the planes is the angle between normals drawn to the planes and the angle between a line and a plane is the compliment of the angle between the line and the normal to the plane 17 APPLICATION OF VECTORS : (a) Work done against a constant force F over a displacement s is defined as W F.s (b) The tangential velocity V of a body moving in a circle is given by V point P w r where r is the pv of the ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) (c) The moment of F about ’O’ is defined as M r Fwhere r is the pv of P wrt ’O’ The direction of M is along the (d) normal to the plane OPN such that r , F & M form a right handed system Moment of the couple = ( r1 r2 ) F where r1 & r2 are pv’s of the point of the application of the forces F & F -D COORDINATE GEOMETRY USEFUL RESULTS A General : (1) Distance (d) between two points (x1 , y1 , z1) and (x2 , y2 , z2) d= (2) ( x x1 ) ( y y1 ) (z z1 ) Section Fomula m x1 m1 x m y1 m1 y ; y= x= m1 m m1 m ( For external division take –ve sign ) ; z= m z1 m1 z m1 m Direction Cosine and direction ratio's of a line (3) (a) Direction cosine of a line has the same meaning as d.c's of a vector Any three numbers a, b, c proportional to the direction cosines are called the direction ratios i.e l m n a b c a b c2 same sign either +ve or –ve should be taken through out note that d.r's of a line joining x1 , y1 , z1 and x2 , y2 , z2 are proportional to x2 – x1 , y2 – y1 and z2 – z1 (b) If is the angle between the two lines whose d.c's are l1 , m1 , n1 and l2 , m2 , n2 cos = l1 l2 + m1 m2 + n1 n2 hence if lines are perpendicular then l1 l2 + m1 m2 + n1 n2 = l1 m1 n1 if lines are parallel then l2 m n l1 m1 n1 note that if three lines are coplanar then l2 l3 m2 m3 n2 = n3 (4) Projection of the join of two points on a line with d.c's l, m, n are l (x2 – x1) + m(y2 – y1) + n(z2 – z1) B (i) (ii) PLANE General equation of degree one in x, y, z i.e ax + by + cz + d = represents a plane Equation of a plane passing through (x1 , y1 , z1) is a (x – x1) + b (y – y1) + c (z – z1) = where a, b, c are the direction ratios of the normal to the plane Equation of a plane if its intercepts on the co-ordinate axes are x1 , y1 , z1 is x y z x1 y1 z1 (iii) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) (iv) Equation of a plane if the length of the perpendicular from the origin on the plane is p and d.c's of the perpendicular as l , m, , n is l x + m y + n z = p Parallel and perpendicular planes – Two planes a1 x + b1 y + c1z + d1 = and a2x + b2y + c2z + d2 = are perpendicular if a1 a2 + b1 b2 + c1 c2 = (v) (vi) parallel if a1 a2 b1 b2 c1 c2 and coincident if a1 a2 b1 b2 c1 c2 d1 d2 Angle between a plane and a line is the compliment of the angle between the normal to the plane and the b.n Line : r a b " then cos(90 ) sin line If Plane : r n d | b | | n | ! where is the angle between the line and normal to the plane (vii) Length of the perpendicular from a point (x1 , y1 , z1) to a plane ax + by + cz + d = is p= (viii) ax1 by1 cz1 d a b2 c2 Distance between two parallel planes ax + by + cz + d1 = and ax + by + cz + d2 = is d1 d a (ix) b c2 Planes bisecting the angle between two planes a1x + b1y + c1z + d1 = and a2 + b2y + c2z + d2 = is given by a1x b1 y c1z d1 a12 b12 c12 = a x b y c2z d (x) a 22 b 22 c 22 Of these two bisecting planes , one bisects the acute and the other obtuse angle between the given planes Equation of a plane through the intersection of two planes P1 and P2 is given by P1 + P2 = C STRAIGHT LINE IN SPACE (i) Equation of a line through A (x1 , y1 , z1) and having direction cosines l ,m , n are x x1 y y1 z z1 l m n and the lines through (x1 , y1 ,z1) and (x2 , y2 ,z2) x x1 y y1 z z1 x x1 y y1 z z1 Intersection of two planes a1x + b1y + c1z + d1 = and a2x + b2y + c2z + d2 = together represent the unsymmetrical form of the straight line x x1 y y1 z z1 General equation of the plane containing the line is l m n A (x – x1) + B(y – y1) + c (z – z1) = where Al + bm + cn = (ii) (iii) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) LINE OF GREATEST SLOPE AB is the line of intersection of G-plane and H is the horizontal plane Line of greatest slope on a given plane, drawn through a given point on the plane, is the line through the point 'P' perpendicular to the line of intersetion of the given plane with any horizontal plane EXERCISE–I Q.1 If a &b are non collinear vectors such that, q ( y 2x 2)a ( x y 1)b , find x & y such that 3p 2q p ( x y )a ( x y 1) b Q.2 (a) (b) Q.3 Points X & Y are taken on the sides QR & RS , respectively of a parallelogram PQRS, so that QX Show that the points a b c ; a b c & b 10 c are collinear Prove that the points A = (1,2,3), B (3,4,7), C ( 3, 2, 5) are collinear & find the ratio in which B divides AC XR 21 PR 25 Find out whether the following pairs of lines are parallel, non-parallel & intersecting, or non-parallel & non-intersecting & RY Q.4 & (i) (iii) YS The line XY cuts the line PR at Z Prove that PZ r1 i r2 2i r1 i r2 2i j 2k j 3i 3k k 6i i 3j 2j 3j 4i 4k 4j (ii) 8k r1 i r2 2i j 3k 4j i 6k j k 2i j 3k 4k j k Q.5 Let OACB be paralelogram with O at the origin & OC a diagonal Let D be the mid point of OA Using vector method prove that BD & CO intersect in the same ratio Determine this ratio Q.6 In a #ABC, points E and F divide sides AC and AB respectively so that AE AF = and = EC FB Suppose D is a point on side BC Let G be the intersection of EF and AD and suppose D is situated so that Q.7 AG BD a = If the ratio = , where a and b are in their lowest form, find the value of (a + b) GD DC b ‘O’is the origin of vectors and A is a fixed point on the circle of radius‘a’with centre O The vector OA is denoted by a A variable point ‘P’ lies on the tangent at A & OP = r Show that if P $ (x,y) & A $ (x1,y1) deduce the equation of tangent at A to this circle Q.8 a r a Hence Let u be a vector on rectangular coordinate system with sloping angle 60° Suppose that u ˆi is geometric mean of u and u 2ˆi where ˆi is the unit vector along x-axis then u has the value equal to a b where a, b N, find the value (a + b)3 + (a – b)3 ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 10 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) The resultant of two vectors a & b is perpendicular to a If b Q.9 a show that the resultant of 2a & b is perpendicular to b Q.10 a , b, c and d are the position vectors of the points A $ (x, y, z) ; B $ (y, – 2z, 3x) ; C $ (2z, 3x, – y) ^ and D$(1,–1, 2) respectively If | a | = ; a b = a^c ; a ^d = and a ^ˆj is obtuse, then find x, y,, z Q.11 If r and s are non zero constant vectors and the scalar b is chosen such that r b s is minimum, then show that the value of b s | r b s |2 is equal to | r |2 Q.12(a) Find a unit vector â which makes an angle ( /4) with axis of z & is such that aˆ ˆi ˆj is a unit vector (b) Prove that a a2 b b2 a b |a| |b| Q.13 Given four non zero vectors a , b , c and d The vectors a , b & c are coplanar but not collinear pair by % % % % then pair and vector d is not coplanar with vectors a , b & c and (a b) (b c) , (da) , (d b) % cos ) prove that ( d c) cos (cos Q.14 Given three points on the xy plane on O(0, 0), A(1, 0) and B(–1, 0) Point P is moving on the plane satisfying the condition P A ·P B + O A ·O B = If the maximum and minimum values of P A P B are M and m respectively then find the value of M2 + m2 Q.15 In the plane of a triangle ABC, squares ACXY, BCWZ are described , in the order given, externally to the triangle on AC & BC respectively Given that CX a.y x.b Q.16 b , CA a , CW x , CB y Prove that Deduce that AW BX Given that u ˆi 2ˆj 3kˆ ; v 2ˆi ˆj 4kˆ ; w ˆi 3ˆj 3kˆ and (u ·R 10)ˆi ( v ·R 20)ˆj ( w ·R 20)kˆ = Find the unknown vector R Question nos 17, 18, 19: Suppose the three vectors a , b, c on a plane satisfy the condition that | a | | b | | c | = | a b | = 1; c is perpendicular to a and b ·c > 0, then Q.17 Find the angle formed by 2a b and b Q.18 If the vector c is expressed as a linear combination Q.19 For real numbers x, y the vector p xa a b then find the ordered pair ( , ) yc satisfies the condition p ·a and p ·b Find the maximum value of p ·c Q.20 (a) (b) If a b c , show that a x b b x c c x a Deduce the Sine rule for a # ABC Find the minimum area of the triangle whose vertices are A(–1, 1, 2); B(1, 2, 3) and C(t, 1, 1) where t is a real number ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 11 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.21 Q.22 (a) Determine vector of magnitude which is perpendicular to both the vectors: 4ˆi ˆj 3kˆ & 2ˆi ˆj 2kˆ (b) A triangle has vertices (1, 1, 1) ; (2, 2, 2), (1, 1, y) and has the area equal to csc Find the value of y sq units Consider a parallelogram ABCD Let M be the centre of line segment BC and S denote the point of intersection of the line segment AM and the diagonal BD Find the ratio of the area of the parallelogram to the area of the triangle BMS Q.23 If a , b , c ,d are position vectors of the vertices of a cyclic quadrilateral ABCD prove that : axb bx d d xa (b a ) (d a ) Q.24 bx c cx d dxb (b c) (d c) The length of the edge of the regular tetrahedron D ABC is 'a' Point E and F are taken on the edges AD and BD respectively such that E divides DA and F divides BD in the ratio 2:1 each Then find the area of triangle CEF Q.25 1ˆ 3ˆ i j and x a (q 3) b , y p a q b If x y , then express p 2 as a function of q, say p = f (q), (p & q 0) and find the intervals of monotonicity of f (q) ˆi ˆj and b Let a EXERCISE–II Q.1 The vector OP = ˆi 2ˆj 2kˆ turns through a right angle, passing through the positive x-axis on the way Find the vector in its new position Q.2 The position vectors of the points A, B, C are respectively (1, 1, 1) ; (1, 1, 2) ; (0, 2, 1) Find a unit vector parallel to the plane determined by ABC & perpendicular to the vector (1, 0, 1) Q.3 Let ( a1 a ) ( a b) (a1 c ) ( b1 a ) (c1 a ) (b1 b) (c1 b) ( b1 c) = and if the vectors (c1 c) & Q.4 (i) ˆi cˆj c kˆ are non coplanar, show that the vectors 2ˆ ˆ ˆ &1 i c1 j c1 k are coplaner Given non zero number x1, x2, x3 ; y1, y2, y3 and z1, z2 and z3 Can the given numbers satisfy ˆi aˆj a kˆ ; ˆi a ˆj a kˆ ; 1 ˆi bˆj b kˆ ; ˆi b ˆj b kˆ and 1 x3 *x1x y1 y z1z ' y = and )x x y y z z3 '(x x1 y3 y1 z z1 z3 If xi > and yi < for all i = 1, 2, and P = (x1, x2, x3) ; Q (y1, y2, y3) and O (0, 0, 0) can the triangle POQ be a right angled triangle? x1 y1 z1 (ii) Q.5 x2 y2 z2 The pv's of the four angular points of a tetrahedron are: A j 2k ; B 3i k ; C 4i 3j 6k & D i j k Find : (i) the perpendicular distance from A to the line BC (ii) the volume of the tetrahedron ABCD (iii) the perpendicular distance from D to the plane ABC (iv) the shortest distance between the lines AB & CD ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 12 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.6 Find the point R in which the line AB cuts the plane CDE where a = ˆi 2ˆj kˆ , b = 2ˆi ˆj 2kˆ , c = ˆj kˆ , d = 2ˆi 2ˆj kˆ & e = 4ˆi ˆj 2kˆ a1ˆi a 2ˆj a 3kˆ ; b b1ˆi b 2ˆj b3kˆ and c c1ˆi c 2ˆj c3kˆ then show that the value of the a ·ˆi a ·ˆj a ·kˆ scalar triple product [ na b nb c nc a ] is (n3 + 1) b ·ˆi b ·ˆj b ·kˆ c ·ˆi c ·ˆj c ·kˆ b a x (a x b ) Q.8(a) Prove that a x b = Q.7 If a p , b q & ( b ) , where µ is a scalar then (b) Given that a,b,p,q are four vectors such that a b prove that ( a q ) p Q.9 ( p q ) a p q ABCD is a tetrahedron with pv's of its angular points as A(–5, 22, 5); B(1, 2, 3); C(4, 3, 2) and D(–1, 2, – 3) If the area of the triangle AEF where the quadrilaterals ABDE and ABCF are parallelograms is S then find the value of S Q.10 If A , B & C are vectors such that | B | | C | , Prove that: A B x A C x Bx C B C Q.11 Given the points P (1, 1, –1), Q (1, 2, 0) and R (–2, 2, 2) Find (a) (b) PQ PR Equation of the plane containing the points P, Q and R (i) in scalar dot product form (ii) in parametric form (iii) in cartesian form and if the plane through PQR cuts the coordinate axes at A, B, C then the area of the #ABC Q.12 Find the scalars & if a x ( b x c) (a b) b (4 sin ) b ( 1) c & ( c c) a c while b & c are non zero non collinear vectors Q.13 Let a i a b Q.14 b c 3k , b i j k and c 2i j j k Find the value(s) of , if any, such that c a = Find the vector product when Find a vector v which is coplanar with the vectors vector i Q.15 2j = &i 2j k It is given that the projection of v along the vector i k and is orthogonal to the j k is equal to If the vectors b , c ,d are not coplanar, then prove that the vector (a b ) ( c d ) (a c) (d b ) (a d ) ( b c ) is parallel to a Q.16 The figure shows non zero vector v, w and z with z orthogonal to the line L, and v and w making equal angles with the line L Assuming | v | = | w | , if the vector w is expressed as a linear combination of v and z as w xv yz Find the value of x and y ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 13 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.17 Consider the non zero vectors a , b , c & d such that no three of which are coplanar then prove that a b cd c a bd b a cd d a b c Hence prove that if a , b , c & d represent the position vectors of the vertices of a plane quadrilateral then Q.18 b cd a bd a cd abc The base vectors a1,a 2,a are given in terms of base vectors b1,b ,b as, a1 a2 b1 2b 2b3 & a 2b1 3b b3 ; 2b1 b 2b If F 3b1 b 2 b3 , then express F in terms of a 1, a & a -1" +0 ; b +, 3! -2 " +1 ; c +,0! -1" + Find the numbers , , & such that +, ! Q.19 Let a Q.20 (a) If px ( x a ) (b) Solve the following equation for the vector p ; pxa b ; (p 0) prove that x p2 b a b &c - 2" + +, ! ( b a ) a p(b xa ) p (p a ) b x c where a , b , c are non p.b c zero non coplanar vectors and a is neither perpendicular to b nor to c , hence show that abc c is perpendicular to b c a ·c p a Q.21 Let a , b & c be non coplanar unit vectors, equally inclined to one another at an angle If axb Q.22 bx c pa qb r c Find scalars p, q & r in terms of Solve the simultaneous vector equations for the vectors x and y x c y a and y c x Q.23 b where c is a non zero vector Consider the points A ( a ); B( b); C( c ) and D (d ) x is the distance of the point A from the plane BCD y is the distance of the point D from the plane ABC Column-I Column-II (A) b c c d d b x [ b c d ] equals (P) [a b c] [ b d c ] [d a c ] (B) a b b c c a y [a b c ] equals (Q) [ a b c ] [ a c d ] [ a d b] (C) [a b c] is equal to when the points A, B, C and D are coplanar (R) [ d a b] [ d b c ] [ d c a ] (D) [d a b] is equal to when the points A, B, C and D are coplanar (S) [ b c d ] [ c a d ] [d a b ] ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 14 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) EXERCISE–III Q.1 Find the angle between the two straight lines whose direction cosines l, m, n are given by 2l + 2m – n = and mn + nl + lm = Q.2 P is any point on the plane lx + my + nz = p A point Q taken on the line OP (where O is the origin) such that OP OQ = p2 Show that the locus of Q is p( lx + my + nz ) = x2 + y2 + z2 Q.3 Find the equation of the plane through the points (2, 2, 1), (1, –2, 3) and parallel to the x-axis Q.4 Through a point P (f, g, h), a plane is drawn at right angles to OP where 'O' is the origin, to meet the r5 coordinate axes in A, B, C Prove that the area of the triangle ABC is where OP = r 2f g h Q.5 The plane lx + my = is rotated about its line of intersection with the plane z = through an angle Prove that the equation to the plane in new position is lx + my + z l m tan =0 Q.6 Find the equations of the straight line passing through the point (1, 2, 3) to intersect the straight line x + = (y – 2) = z + and parallel to the plane x + 5y + 4z = Q.7 Find the equations of the two lines through the origin which intersect the line angle of Q.8 Q.9 x y z at an 1 A variable plane is at a constant distance p from the origin and meets the coordinate axes in points A, B and C respectively Through these points, planes are drawn parallel to the coordinates planes Find the locus of their point of intersection x 2 y 3z Find the distance of the point P (– 2, 3, – 4) from the line measured parallel to the plane 4x + 12y – 3z + = Q.10 Find the equation to the line passing through the point (1, –2, –3) and parallel to the line 2x + 3y – 3z + = = 3x – 4y + 2z – Q.11 Find the equation of the line passing through the point (4, –14, 4) and intersecting the line of intersection of the planes : 3x + 2y – z = and x – 2y – 2z = –1 at right angles Q.12 Let P = (1, 0, – 1) ; Q = (1, 1, 1) and R = (2, 1, 3) are three points (a) Find the area of the triangle having P, Q and R as its vertices (b) Give the equation of the plane through P, Q and R in the form ax + by + cz = (c) Where does the plane in part (b) intersect the y-axis (d) Give parametric equations for the line through R that is perpendicular to the plane in part (b) Q.13 Find the point where the line of intersection of the planes x – 2y + z = l and x + 2y – 2z = 5, intersects the plane 2x + 2y + z + = Q.14 Feet of the perpendicular drawn from the point P (2, 3, –5) on the axes of coordinates are A, B and C Find the equation of the plane passing through their feet and the area of #ABC Q.15 Find the equations to the line which can be drawn from the point (2, –1, 3) perpendicular to the lines x y z x y z and at right angles 4 x y z Find the equation of the plane containing the straight line and perpendicular to the plane x – y + z + = Q.16 ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 15 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.17 Find the value of p so that the lines x y p z x and 1 y z are in the same plane For this value of p, find the coordinates of their point of intersection and the equation of the plane containing them Q.18 Q.19 Find the equations to the line of greatest slope through the point (7, , –1) in the plane x – 2y + 3z = assuming that the axes are so placed that the plane 2x + 3y – 4z = is horizontal -2" - 1" Let L be the line given by r = + + + and let P be the point (2, – 1, 1) Also suppose that E be +, ! ,+ 1! the plane containing three non collinear points A = (0, 1, 1); B(1, 2, 2) and C = (1, 0, 1) Find Distance between the point P and the line L Equation of the plane E Equation the plane F containing the line L and the point P Acute between the plane E and F Volume of the parallelopiped by A, B, C and the point D(– 3, 0, 1) (a) (b) (c) (d) (e) Q.20 The position vectors of the four angular points of a tetrahedron OABC are (0, 0, 0); (0, 0, 2); (0, 4, 0) and (6, 0, 0) respectively A point P inside the tetrahedron is at the same distance 'r' from the four plane faces of the tetrahedron Find the value of 'r' Q.21 The line x y 10 z 14 is the hypotenuse of an isosceles right angled triangle whose opposite vertex is (7, 2, 4) Find the equation of the remaining sides Q.22 Q.23 Q.24 Find the foot and hence the length of the perpendicular from the point (5, 7, 3) to the line x 15 y 29 z Also find the equation of the plane in which the perpendicular and the given straight line lie x y z Find the equation of the line which is reflection of the line in the plane 3x – 3y + 10z = 26 Find the equation of the plane containing the line x y z x and parallel to the line 2 y z Find also the S.D between the two lines Q.25 Consider the plane E: - 1" r = +1 + +, ! -1" +2 + +, ! -1" +0 +, ! Let F be the plane containing the point A (– 4, 2, 2) and parallel to E Suppose the point B is on the plane E such that B has a minimum distance from the point A If C (– 3, 0, 4) lies in the plane F Find the area of the angle ABC ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 16 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) EXERCISE–IV Q.1(a) Select the correct alternative : (i) If the vectors a , b & c form the sides BC, CA & AB respectively of a triangle ABC, then (A) a b b.c c.a = (B) a b b c c a (C) a b b.c c.a (D) a b b c c a =0 (ii) Let the vectors a , b , c & d be such that a b c d = Let P1 & P2 be planes determined by the pairs of vectors a , b & c , d respectively Then the angle between P1 and P2 is : (A) (B) /4 (C) /3 (4) /2 (iii) If a , b & c are unit coplanar vectors, then the scalar triple product 2a b 2b (A) a = c 2c (B) (C) (D) 3 [ JEE ,2000 (Screening) + + out of 35 ] Let ABC and PQR be any two triangles in the same plane Assume that the perpendiculars from the points A, B, C to the sides QR, RP, PQ respectively are concurrent Using vector methods or otherwise, prove that the perpendiculars from P, Q, R to BC, CA, AB respectively are also concurrent [ JEE '2000 (Mains) 10 out of 100 ] (b) Q.2(i) If a = i j k, b = a + b and b i 2k & c = 2j i k , find a unit vector normal to the vectors 2j c (ii) Given that vectors a & b are perpendicular to each other, find vector v in terms of a & b satisfying the equations, v ·a = , b = and [ v a b ] = (iii) a , b & c are three unit vectors such that a b a & b given that vectors b & c are non-parallel c = c Find angle between vectors b [ REE '2000 (Mains) + + out of 100] Q.3(a) The diagonals of a parallelogram are given by vectors i j k and 3i and also the area (b) Find the value of such that a, b, c are all non-zero and i j a (3i j k)b ( i j 3k)c = (ai Q.4(a) Find the vector r which is perpendicular to a = i (b) Two vertices of a triangle are at vector of third vertex i j and i Q.5(a) If a , b and c are unit vectors, then a (A) (b) Let a ˆi kˆ , b (A) only x Q.6 (B) x ˆi ˆj (1 x )kˆ and c (B) only y Let A(t ) = f1 (t ) i f2 (t ) j and B( t ) b bj ck ) j 5k and b 4j [ REE '2001 (Mains) + 3] 2i j k and r 2ˆi ˆj kˆ +8=0 j and its orthocentre is at i j Find the position [ REE '2001 (Mains) + 3] b c 2 c a does NOT exceed (C) (D) yˆi x ˆj (1 x y) kˆ Then [a , b, c] depends on (C) NEITHER x NOR y (D) both x and y [ JEE '2001 (Screening) + out of 35] g1 ( t ) i g (t ) j , t [0, 1], where f1, f2, g1, g2 are continuous functions If A(t ) and B( t ) are nonzero vectors for all t and A(0) = i B(0) = 3i j and B(1) = i k Determine its sides j , A(1) = i 2j, j , then show that A(t ) and B(t ) are parallel for some t [JEE '2001 (Mains) out of 100] ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 17 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.7(a) If a and b are two unit vectors such that a + b and a – b are perpendicular to each other then the angle between a and b is (A) 450 (B) 600 (C) cos–1 (D) cos–1 (b) Let V 2ˆi ˆj kˆ and W ˆi 3kˆ If U is a unit vector, then the maximum value of the scalar triple product U V W is (A) –1 Q.8 (B) 10 (D) 60 [JEE 2002(Screening), + 3] Let V be the volume of the parallelopiped formed by the vectors a a1ˆi a ˆj a 3kˆ , b b1ˆi b ˆj b3 kˆ , c c1ˆi (C) 59 c ˆj c3 kˆ If ar , br , cr , where r = 1, 2, 3, are non-negative real numbers and / ar b r c r = 3L, show that V L3 [JEE 2002(Mains), 5] r Q.9 Q.10 If a = ˆi aˆj kˆ , b = ˆj akˆ , c = aˆi kˆ , then find the value of ‘a’ for which volume of parallelopiped formed by three vectors as coterminous edges, is minimum, is 1 (A) (B) – (C) ± (D) none 3 [JEE 2003(Scr.), 3] (i) Find the equation of the plane passing through the points (2, 1, 0) , (5, 0, 1) and (4, 1, 1) (ii) If P is the point (2, 1, 6) then find the point Q such that PQ is perpendicular to the plane in (i) and the mid point of PQ lies on it [JEE 2003, out of 60] Q.11 If u , v , w are three non-coplanar unit vectors and , , & are the angles between u and v , v and w , w and u respectively and x , y, z are unit vectors along the bisectors of the angles , , & & u v w sec sec sec respectively Prove that x y y z z x 16 2 [JEE 2003, out of 60] y z x y k z and intersect, then k = (A) 2/9 (B) 9/2 (C) (D) – (b) A unit vector in the plane of the vectors 2ˆi ˆj kˆ , ˆi ˆj kˆ and orthogonal to 5ˆi 2ˆj 6kˆ 3ˆj kˆ 6ˆi 5kˆ 2ˆi 5kˆ 2ˆi ˆj kˆ (A) (C) (D) (B) 61 10 29 (c) If a ˆi j kˆ , a ·b and a b ˆj kˆ , then b = (A) ˆi (B) ˆi ˆj kˆ (C) 2ˆj kˆ (D) 2ˆi Q.12(a) If the lines x [ JEE 2004 (screening)] Q.13(a) Let a , b, c, d are four distinct vectors satisfying a b = c d and a c b d Show that a · b c ·d a ·c b ·d (b)Let P be the plane passing through (1, 1, 1) and parallel to the lines L1 and L2 having direction ratios 1, 0, –1 and –1, 1, respectively If A, B and C are the points at which P intersects the coordinate axes, find the volume of the tetrahedron whose vertices are A, B, C and the origin [JEE 2004, + 2out of 60] ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 18 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.14(a) A variable plane at a distance of unit from the origin cuts the co-ordinate axes at A, B and C If the 1 centroid D (x, y, z) of triangle ABC satisfies the relation = k, then the value of k is x y z2 (A) (B) (C) 1/3 (D) [JEE 2005 (Scr), 3] (b) Find the equation of the plane containing the line 2x – y + z – = 0, 3x + y + z = and at a distance of from the point (2, 1, – 1) (c) Incident ray is along the unit vector vˆ and the reflected ray is along the ˆ The normal is along unit vector aˆ outwards Express unit vector w ˆ in terms of aˆ and vˆ w [ JEE 2005 (Mains), + out of 60] Q.15(a) A plane passes through (1, –2, 1) and is perpendicular to two planes 2x – 2y + z = and x – y + 2z = The distance of the plane from the point (1, 2, 2) is (A) (B) (C) (D) 2 (b) Let a ˆi 2ˆj kˆ , b ˆi ˆj kˆ and c ˆi ˆj kˆ A vector in the plane of a and b whose projection on c has the magnitude equal to , is (A) 4ˆi ˆj kˆ (B) 3ˆi ˆj 3kˆ (C) 2ˆi ˆj 2kˆ (D) 4ˆi ˆj kˆ [JEE 2006,3 marks each] (c) Let A be vector parallel to line of intersection of planes P1 and P2 through origin P1 is parallel to the vectors ˆj + kˆ and ˆj – kˆ and P2 is parallel to ˆj – kˆ and ˆi + ˆj , then the angle between vector A and ˆi + ˆj – kˆ is (A) (B) (C) (d) Match the following (i) Two rays in the first quadrant x + y = | a | and ax – y = intersects each other in the interval a (a0, 0), the value of a0 is (ii) Point ( , , &) lies on the plane x + y + z = ˆi ˆj & kˆ , kˆ (kˆ a ) = 0, then & = Let a (iii) (1 (D) (A) (B) 4/3 (y y ) dy + [JEE 2006, 5] 1) dy (C) 1 x dx + x dx (iv) In a #ABC, if sinA sinB sinC + cos A cosB = 1, then the value of sin C = (D) [JEE 2006, 6] (e) Match the following (i) / tan i 1 2i t , then tan t = (A) (ii) Sides a, b, c of a triangle ABC are in A.P and cos a , cos b , cos (B) c , then tan tan 3 b c a c a b 2 (iii) A line is perpendicular to x + 2y + 2z = and passes through (0, 1, 0) The perpendicular distance of this line from the origin is = (D) 2/3 [JEE 2006, 6] (C) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 19 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.16(a) The number of distinct real values of , for which the vectors are coplanar, is (A) zero (B) one (C) two (b) Let a , b, c be unit vectors such that a (A) a b b c c a (C) a b b c a c 2ˆ i ˆj kˆ , ˆi 2ˆ j kˆ and ˆi ˆj 2ˆ k (D) three b c Which one of the following is correct? (B) a b b c c a (D) a b, b c, c a are mutually perpendicular (c) Let the vectors P Q , Q R , R S , S T , T U and U P represent the sides of a regular hexagon Statement-1: P Q × R S S T because Statement-2: P Q R S = and P Q S T (A) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for statement-1 (B) Statement-1 is true, statement-2 is true; statement-2 is NOT a correct explanation for statement-1 (C) Statement-1 is true, statement-2 is false (D) Statement-1 is false, statement-2 is true (d) Consider the planes 3x – 6y – 2z = 15 and 2x + y – 2z = Statement-1: The parametric equations of the line of intersection of the given planes are x = + 14t, y = + 2t, z = 15t because Statement-2: The vector 14ˆi 2ˆj 15kˆ is parallel to the line of intersection of given planes (A) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for statement-1 (B) Statement-1 is true, statement-2 is true; statement-2 is NOT a correct explanation for statement-1 (C) Statement-1 is true, statement-2 is false (D) Statement-1 is false, statement-2 is true MATCH THE COLUMN: (e) Consider the following linear equations ax + by + cz = bx + cy + az = cx + ay + bz = Match the conditions/ expressions in Column I with statements in Column II Column I Column II (A) a + b + c and (P) the equation represent planes 2 a + b + c = ab + bc + ca meeting only at a single point (B) a + b + c = and (Q) the equation represent the line 2 a + b + c ab + bc + ca x=y=z (C) a + b + c and (R) the equation represent identical planes 2 a + b + c ab + bc + ca (D) a + b + c = and (S) the equation represent the whole of 2 = a + b + c ab + bc + ca the three dimensional space [JEE 2007, 3+3+3+3+6] ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 20 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.17(a) The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors aˆ , bˆ, cˆ such that aˆ ·bˆ (A) bˆ ·cˆ cˆ ·aˆ (B) Then the volume of the parallelopiped is (C) 2 (D) (b) Let two non-collinear unit vector aˆ and bˆ form an acute angle A point P moves so that at any time t the position vector O P (where O is the origin) is given by aˆ cos t bˆ sin t When P is farthest from origin O, let M be the length of O P and uˆ be the unit vector along O P Then, (A) uˆ aˆ bˆ and M | aˆ bˆ | (1 aˆ ·bˆ) (C) uˆ aˆ bˆ and M | aˆ bˆ | (1 2aˆ ·bˆ) (B) uˆ aˆ bˆ and M | aˆ bˆ | (1 aˆ ·bˆ) (D) uˆ aˆ bˆ and M | aˆ bˆ | (1 2aˆ ·bˆ) 1 (c) Consider three planes P1 : x – y + z = P2 : x + y – z = –1 P3 : x – 3y + 3z = Let L1, L2, L3 be the lines of intersection of the planes P2 and P3, P3 and P1, and P1 and P2, respectively Statement-1 : At least two of the lines L1, L2 and L3 are non-parallel and Statement-2 : The three planes not have a common point (A) Statement-1 is True, Statement-2 is True; statement-2 is a correct explanation for statement-1 (B) Statement-1 is True, Statement-2 is True; statement-2 is NOT a correct explanation for statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True Paragraph for Question Nos (i) to (iii) (d) Consider the lines x y z x ; L2 : The unit vector perpendicular to both L1 and L2 is L1 : (i) ˆi 7ˆj 7kˆ ˆi 7ˆj 5kˆ (B) 99 The shortest distance between L1 and L2 is (A) (ii) z 3 ˆi 7ˆj 5kˆ (D) 7ˆi 7ˆj kˆ 99 17 41 17 (C) (D) 5 The distance of the point (1, 1, 1) from the plane passing through the point (–1, –2, –1) and whose normal is perpendicular to both the lines L1 and L2 is (A) (iii) (C) y 2 (A) (B) 75 (B) 75 (C) 13 75 23 75 [JEE 2008, 3+3+3+4+4+4] (D) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 21 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.18(a) Let P(3, 2, 6) be a point in space and Q be a point on the line r (ˆi ˆj 2kˆ ) ( 3ˆi ˆj 5kˆ ) Then the value of for which the vector PQ is parallel to the plane x – 4y + 3z =1 is (A) (B) – (C) (D) – (b) If a , b, c and d are unit vectors such that a b · c d and a ·c , then (A) a , b, c are non-coplanar (B) b, c, d are non-coplanar (C) b, d are non-parallel (D) a , d are parallel and b, c are parallel (c) A line with positive direction cosines passes through the point P(2, – 1, 2) and makes equal angles with the coordinate axes The line meets the plane 2x + y + z = at point Q The length of the line segment PQ equals (A) (B) (C) (D) (d) Match the statements/expressions given in Column I with the values given in Column II Column I Column II (A) Root(s) of the equation sin2 + sin22 = (P) (B) - 6x " - 3x " Points of discontinuity of the function f(x) = + cos + , , ! , ! (Q) where [y] denotes the largest integer less than or equal to y (C) Volume of the parallelopiped with its edges represented by the vectors (R) ˆi ˆj, ˆi 2ˆj and ˆi ˆj (D) kˆ Angle between vectors a and b where a , b and c are unit (S) vectors satisfying a b (T) 3c (e) Let (x, y, z) be points with integer coordinates satisfying the system of homogeneous equations : 3x – y – z = – 3x + z = – 3x + 2y + z = Then the number of such points for which x2 + y2 + z2 100 is [JEE 2009, 3+3+3+8+4] ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 22 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) ANSWER KEY EXERCISE–I Q.1 x = 2, y = Q.4 Q.5 (i) parallel (ii) the lines intersect at the point p.v i j : Q.6 Q.7 xx1 + yy1 = a2 Q.8 Q.12 (a) Q.17 i j k Q.14 34 , 3 Q.18 (b) externally in the ratio : Q.2 (a) ± 3( ˆi 2ˆj kˆ ), (b) y = or y = – Q.22 12 Q.24 5a 12 Q.20 (b) Q.18 sq units 28 x = 2, y = – 2, z = – Q.10 ˆi 2ˆj 5kˆ Q.16 Q.19 (iii) lines are skew Q.25 p = q(q 3) ; decreasing in q (–1, 1), q EXERCISE–II ˆ i Q.1 ˆ j ˆ k (i) Q.6 p.v of R = r = 3i + 3k (a) 2ˆi 3ˆj 3kˆ , (b) (i) r ·n area = Q.14 ( 1) n ,n p= or Q.22 x I& k = – 1, Q.19 Q.21 j (ˆi 5ˆj kˆ ) Q.4 NO, NO 110 Q.9 ˆi ˆj kˆ + , (ii) rˆ (ˆj kˆ ) ( 3ˆi ˆj 3kˆ ) ,(iii) 2x–3y+3z+4=0, 22 n Q.12 3 14 (ii) (iii) 10 (iv) Q.5 Q.11 Q.2 Q.16 = – 2, & = 1 p= cos ; q= 1 cos | v | sin |z| x = and y = Q.20 cos = 2/3 ; if = then vector product is 60 i Q.13 cos ; q= * (b) )p ( ; r= cos 2cos a ( c a ) c b c b (c b) c a c , y c c2 Q.18 abc a c b a c a b k F = 2a1 5a 3a b.c b a b b b c a b 1 ; r= cos 1 cos Q.23 (A) Q ; (B) R ; (C) S ; (D) P ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 23 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) EXERCISE–III Q.1 = 900 y + 2z = Q.3 Q.7 x y z Q.10 x Q.12 (a) Q.13 (1, –2, – 4) Q.16 2x + 3y + z + = Q.19 (a) Q.21 x Q.22 (9, 13, 15) ; 14 ; 9x – 4y – z = 14 Q.24 x – 2y + 2z – = 0; units y 13 2x ; (b) x or y z z 17 2y Q.14 Q.6 x Q.8 x2 Q.11 x y 2 y2 z2 z 3 p2 y 14 10 17 z 4 z = 1; (c) 0, , ; (d) x = 2t + ; y = 2t + and z = – t + 3 x y z 19 , Area = sq units Q.17 p = 3, (2, 1, –3) ; x + y + z = ; (b) x + y – 2z + = 0; (c) x – 2y + z = 5; (d) /3; (e) y Q.9 z x ; 2 y Q.15 x 11 y 10 z Q.18 x 22 y z Q.20 z x Q.25 9/2 Q.23 y z EXERCISE–IV Q.3 (a) Q.4 (a) r Q.7 (a) B ; (b) C Q.12 (a) B, (b) B, (c) A Q.13 (b) 9/2 cubic units Q.14 (a) D; (b) 2x – y + z – = and 62x + 29y + 19z – 105 = 0, (c) wˆ = vˆ – 2( aˆ · vˆ ) aˆ Q.15 (a) D; (b) A; (c) B ; (d) (i) D, (ii) A, (iii) B, C, (iv) D; (e) (i) B, (ii) D, (iii) C Q.16 (a) C; (b) B; (c) C; (d) D; (e) (A) R; (B) Q; (C) P; (D) S Q.17 (a) A; (b) A; (c) D; (d) (i) B; Q.18 (a) A ; (b) C ; (c) C ; (d) (A) Q, S ; (B) P, R, S, T ; (C) T ; (D) R ; (e) j 7k , Q.2 (i) + i ; (ii) a xb ; (iii) (a b ) (a) (i) B 5i (ii) A (iii) A b b2 Q.1 1 ( i j 5k); 1274 sq units (b) 2 13i 11 j k ; (b) Q.9 ˆ 17 ˆ i j 7 D Q.5 = 0, = –2 + 29 (a) B (b) C Q.10 (i) x + y – 2z = ; (ii) (6, 5, –2) (ii) D; (iii) C ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 24 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005)