4.5. COORDINATES,VECTORS,CURVES, AND SURFACES IN SPACE 115 r X x y O Y Z z Mxyz(,,) M M M X Y Z Figure 4.29. Point in rectangular Cartesian coordinate system. of intersection of the perpendiculars with the axes by M X , M Y ,andM Z , respectively. The numbers (see Fig. 4.29) x = OM X , y = OM Y , z = OM Z ,(4.5.2.1) where OM X , OM Y ,andOM Z are the signed lengths of the directed segments −−→ OM X , −−→ OM Y ,and −−→ OM Z of the axes OX, OY ,andOZ, respectively, are called the coordinates of the point M in the rectangular Cartesian coordinate system. The number x is called the first coordinate or the abscissa of the point M, the number y is called the second coordinate or the ordinate of the point M, and the number z is called the third coordinate or the applicate of the point M. Usually one says that the point M has the coordinates (x, y, z), and the notation M(x, y, z)isused. To each point M of three-dimensional space one can assign its position vector. The directed segment −−→ OM is called the position vector of the point M . The position vector determines the vector r (r = −−→ OM) whose coordinates are its projections on the axes OX, OY,andOZ, respectively. Obviously, the triple (x, y, z) of numbers can be called the point M whose coordinates are these numbers, or the position vector −−→ OM whose projections are these numbers. An arbitrary vector (x, y, z) can be represented as (x , y, z)=xi + yj + zk,(4.5.2.2) where i =(1, 0, 0), j =(0, 1, 0), and k =(0, 0, 1) are the unit vectors with the same directions as the coordinate axes OX, OY ,andOZ. The distance between points M 1 (x 1 , y 1 , z 1 )and M 2 (x 2 , y 2 , z 2 ) is determined by the formula d = (x 2 – x 1 ) 2 +(y 2 – y 1 ) 2 +(z 2 – z 1 ) 2 = |r 2 – r 1 |,(4.5.2.3) where r 2 = −−→ OM 2 and r 1 = −−→ OM 1 are the position vectors of the points M 1 and M 2 , respectively (see Fig. 4.30). r rr r 1 2 1 21 2 M M X Y Z O Figure 4.30. Distance between points. 116 ANALYTIC GEOMETRY Two vectors a =(x 1 , y 1 , z 1 )andb =(x 2 , y 2 , z 2 ) are equal to each other if and only if the following relations hold simultaneously: x 1 = x 2 , y 1 = y 2 , z 1 = z 2 . For arbitrary vectors, one has the relations (x 1 , y 1 , z 1 ) (x 2 , y 2 , z 2 )=(x 1 x 2 , y 1 y 2 , z 1 z 2 ), α(x, y, z)=(αx, αy, αz). (4.5.2.4) If a point M divides the directed segment −−−→ M 1 M 2 in the ratio p/q =λ, then the coordinates of this point are given by the formulas x = x 1 + λx 2 1 + λ = qx 1 + px 2 q + p , y = y 1 + λy 2 1 + λ = qy 1 + py 2 q + p , z = z 1 + λz 2 1 + λ = qz 1 + pz 2 q + p , or r = r 1 + λr 2 1 + λ = qr 1 + pr 2 q + p ,(4.5.2.5) where –∞ ≤ λ ≤ ∞ . In the special case of the midpoint of the segment −−−→ M 1 M 2 (p = q, λ = 1), the coordinates are x = x 1 + x 2 2 , y = y 1 + y 2 2 , z = z 1 + z 2 2 or r = r 1 + r 2 2 ; i.e., each coordinate of the midpoint of a segment is equal to the half-sum of the corre- sponding coordinates of its endpoints. The angles α, β,andγ between the segment −−−→ M 1 M 2 and the coordinate axes OX, OY , and OZ are determined by the expressions cos α = x 2 – x 1 |r 2 – r 1 | ,cosβ = y 2 – y 1 |r 2 – r 1 | ,cosγ = z 2 – z 1 |r 2 – r 1 | ,(4.5.2.6) and cos 2 α +cos 2 β +cos 2 γ = 1. The numbers cos α,cosβ,andcosγ are called the direction cosines of the segment −−−→ M 1 M 2 . The angle ϕ between arbitrary directed segments −−−→ M 1 M 2 and −−−→ M 3 M 4 joining the points M 1 (x 1 , y 1 , z 1 ), M 2 (x 2 , y 2 , z 2 )andM 3 (x 3 , y 3 , z 3 ), M 4 (x 4 , y 4 , z 4 ), respectively, can be found from the relation cos ϕ = (x 2 – x 1 )(x 4 – x 3 )+(y 2 – y 1 )(y 4 – y 3 )+(z 2 – z 1 )(z 4 – z 3 ) |r 2 – r 1 ||r 4 – r 3 | .(4.5.2.7) The area of the triangle with vertices M 1 , M 2 ,andM 3 is given by the formula S = 1 4 y 1 z 1 1 y 2 z 2 1 y 3 z 3 1 2 + z 1 x 1 1 z 2 x 2 1 z 3 x 3 1 2 + x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 2 .(4.5.2.8) 4.5. COORDINATES,VECTORS,CURVES, AND SURFACES IN SPACE 117 The volume of the pyramid with vertices M 1 , M 2 , M 3 ,andM 4 is equal to V = 1 6 x 2 – x 1 y 2 – y 1 z 2 – z 1 x 3 – x 1 y 3 – y 1 z 3 – z 1 x 4 – x 1 y 4 – y 1 z 4 – z 1 = 1 6 1 x 1 y 1 z 1 1 x 2 y 2 z 2 1 x 3 y 3 z 3 1 x 4 y 4 z 4 ,(4.5.2.9) and the volume of the parallelepiped spanned by vectors −−−→ M 1 M 2 , −−−→ M 1 M 3 ,and −−−→ M 1 M 4 is equal to V = x 2 – x 1 y 2 – y 1 z 2 – z 1 x 3 – x 1 y 3 – y 1 z 3 – z 1 x 4 – x 1 y 4 – y 1 z 4 – z 1 = 1 x 1 y 1 z 1 1 x 2 y 2 z 2 1 x 3 y 3 z 3 1 x 4 y 4 z 4 .(4.5.2.10) The coordinate surfaces, on which one of the coordinates is constant, are planes parallel to the coordinate planes, and the coordinate lines, along which only one coordinate varies, are straight lines parallel to the coordinate axes. Coordinate surfaces meet in the coordinate lines. 4.5.2-2. Transformation of Cartesian coordinates under parallel translation of axes. Suppose that two rectangular Cartesian coordinate systems OXY Z and O X Y Z are given and the first system can be made to coincide with the second system by translating the origin O of the first system to the origin O of the second system. Under this translation the axes preserve their direction (the respective axes of the systems are parallel), and the origin moves by x 0 in the direction of the axis OX,byy 0 in the direction of the axis OY , and by z 0 in the direction of the axis OZ. Obviously, the point O in the coordinate system OXY Z has the coordinates (x 0 , y 0 , z 0 ). An arbitrary point M has coordinates (x, y, z) in the system OXY Z and coordinates (ˆx,ˆy,ˆz) in the system O X Y Z. The transformation of rectangular Cartesian coordinates by the parallel translation of axes is determined by the formulas x =ˆx + x 0 , y =ˆy + y 0 , z =ˆz + z 0 or ˆx = x – x 0 , ˆy = y – y 0 , ˆz = z – z 0 . (4.5.2.11) 4.5.2-3. Transformation of Cartesian coordinates under rotation of axes. Suppose that two rectangular Cartesian coordinate systems OXY Z and O X Y Z are given and the first system can be made to coincide with the second system by rotating the first system around the point O. An arbitrary point M has coordinates (x, y, z) in the system OXY Z and coordinates (ˆx,ˆy,ˆz) in the system O X Y Z.IftheaxisO X has the direction cosines e 11 , e 21 , e 31 ,the axis O Y has the direction cosines e 12 , e 22 , e 32 , and the axis O Z has the direction cosines e 13 , e 23 , e 33 in the coordinate system OXY Z, then the axis OX has the direction cosines e 11 , e 12 , e 13 ,theaxisOY has the direction cosines e 21 , e 22 , e 23 , and the axis OZ has the 118 ANALYTIC GEOMETRY direction cosines e 31 , e 32 , e 33 in the coordinate system O X Y Z. The transformation of rectangular Cartesian coordinates by the rotation of axes is determined by the formulas x = e 11 ˆx + e 12 ˆy + e 13 ˆz, y = e 21 ˆx + e 22 ˆy + e 23 ˆz, z = e 31 ˆx + e 32 ˆy + e 33 ˆz, or ˆx = e 11 x + e 21 y + e 31 z, ˆy = e 12 x + e 22 y + e 32 z, ˆz = e 13 x + e 23 y + e 33 z. (4.5.2.12) 4.5.2-4. Transformation of coordinates under translation and rotation of axes. Suppose that two rectangular Cartesian coordinate systems OXY Z and O X Y Z are given and the first system can be made to coincide with the second system by translating the origin O(0, 0, 0)ofthefirst system to the origin O(x 0 , y 0 , z 0 ) of the second system, and then by rotating the first system around the point O (see Paragraphs 4.5.1-2 and 4.5.1-3). An arbitrary point M has coordinates (x, y, z) in the system OXY Z and coordinates (ˆx,ˆy,ˆz) in the system O X Y Z. The transformation of rectangular Cartesian coordinates by the parallel translation and the rotation of axes is determined by the formulas x=e 11 ˆx + e 12 ˆy + e 13 ˆz + x 0 , y =e 21 ˆx + e 22 ˆy + e 23 ˆz + y 0 , z =e 31 ˆx + e 32 ˆy + e 33 ˆz + z 0 , or ˆx=e 11 (x – x 0 )+e 21 (y – y 0 )+e 31 (z – z 0 ), ˆy=e 12 (x – x 0 )+e 22 (y – y 0 )+e 32 (z – z 0 ), ˆz=e 13 (x – x 0 )+e 23 (y – y 0 )+e 33 (z – z 0 ). (4.5.2.13) 4.5.2-5. Cylindrical and spherical coordinates. A more general curvilinear coordinate system is obtained if one introduces three families of coordinate surfaces such that exactly one surface of each family passes through each point of space. The position of a point in such a system is determined by the values of the parameters of the coordinate surfaces passing through this point. The most commonly used curvilinear coordinate systems (cylindrical and spherical) are described below. The cylindrical coordinates of a point M are defined as the polar coordinates ρ and ϕ (see Paragraph 4.1.2-5) of the projection of M onto the base plane (usually OXY )andthe distance (usually z) from M to the base plane, which is called the applicate (see Fig. 4.31a). To be definite, one usually assumes that 0 <ϕ ≤2π or –π < ϕ≤ π. For cylindrical coordinates, the coordinate surfaces are the planes z = const perpendicular to the axis OZ, the half-planes ϕ = const bounded by the axis OZ, and the cylindrical surfaces ρ = const with axis OZ. The coordinate surfaces intersect in the coordinate lines. r X φφ θ ρ OO X YY Z z ()a ()b Z Mxyz(,,) Mxyz(,,) Figure 4.31. Point in cylindrical (a) and spherical (b) coordinates. 4.5. COORDINATES,VECTORS,CURVES, AND SURFACES IN SPACE 119 The spherical (polar) coordinates are defined as the length r = | −−→ OM| of the radius vector, the longitude ϕ, and the polar distance θ, called the latitude (see Fig. 4.31b). To be definite, one usually assumes that 0 < ϕ ≤ 2π, 0 ≤ θ ≤ π or –π < ϕ ≤ π,and0 ≤ θ ≤ π. For spherical coordinates, the coordinate surfaces are the spheres r = const centered at the origin, the half-planes ϕ = const bounded by the axis OZ, and the cones θ = const with vertex O and axis OZ. The coordinate surfaces intersect in the coordinate lines. 4.5.2-6. Relationship between Cartesian, cylindrical, and spherical coordinates. Let M be an arbitrary point in space with rectangular Cartesian coordinates (x, y, z), cylin- drical coordinates (ρ, ϕ, z), and spherical coordinates (r, ϕ, θ). The formulas of transition from the cylindrical coordinate system to the Cartesian coordinate system and vice versa have the form x = ρ cos ϕ, y = ρ sin ϕ, z = z, or ρ = x 2 + y 2 , tan ϕ = y/x, z = z, (4.5.2.14) where the polar angle ϕ is taken with regard to the quadrant in which the projection of the point M onto the base plane lies. The formulas of transition from the spherical coordinate system to the Cartesian coordinate system and vice versa have the form x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ, or r = x 2 + y 2 + z 2 , tan ϕ = y/x, tan θ = x 2 + y 2 /z, (4.5.2.15) where the angle ϕ is determined from the same considerations as in the case of cylindrical coordinates. 4.5.2-7. Surfaces and curves in space. A surface in space determined by an equation in some coordinate system is the locus of points in space whose coordinates satisfy this equation. An equation of a surface in space in a given coordinate system is an equation with three variables satisfied by the coordinates of points lying on the surface and not satisfied by the coordinates of points that do not lie on the surface. The coordinates of an arbitrary point of the surface occurring in the equation of the surface are called the current coordinates. Example 1. The equation (x – x 0 ) 2 +(y – y 0 ) 2 +(z – z 0 ) 2 = r 2 defines the sphere of radius r centered at the point (x 0 , y 0 , z 0 ), i.e., the locus of points lying at the distance r from the point (x 0 , y 0 , z 0 ). Example 2. The equation x 2 + y 2 +(z – 1) 2 = 0 determines the single point with coordinates (0, 0, 1). Example 3. The equation x 2 + y 2 + z 2 + 1 = 0 does not have solutions for any real x, y,andz. In the general case, the equation of a surface in the Cartesian coordinate system OXY Z can be written as F (x, y, z)=0.(4.5.2.16) 120 ANALYTIC GEOMETRY On the other hand, a surface (continuous surface) can be defined parametrically; i.e., a surface is defined as the set of points whose coordinates satisfy the system of parametric equations x = x(u, v), y = y(u, v), z = z(u, v)(4.5.2.17) for appropriate values of the parameters u and v. In spatial analytic geometry, each curve is treated as the intersection of two surfaces and hence is defined by a system of two equations F (x, y, z)=0, G(x, y, z)=0.(4.5.2.18) On the other hand, each curve (continuous curve) can be defined parametrically; i.e., a curve is defined as the set of points whose coordinates satisfy the system of parametric equations x = x(t), y = y(t), z = z(t)orr = r (t)(–∞ ≤ t 1 ≤ t ≤ t 2 ≤ ∞). (4.5.2.19) 4.5.3. Vectors. Products of Vectors 4.5.3-1. Scalar product of two vectors. The scalar product of two vectors is defined as the product of their absolute values times the cosine of the angle between the vectors (see Fig. 4.32), a ⋅ b = |a||b| cos ϕ.(4.5.3.1) If the angle between vectors a and b is acute, then a ⋅ b > 0; if the angle is obtuse, then a ⋅ b < 0; if the angle is right, then a ⋅ b = 0. Taking into account (4.5.1.1), we can write the scalar product as a ⋅ b = |a||b| cos ϕ = |a| pr a b = |b| pr b a.(4.5.3.2) Remark. The scalar product of a vector a by a vector b is also denoted by (a, b)orab. a b φ Figure 4.32. Scalar product of two vectors. The angle ϕ between vectors is determined by the formula cos ϕ = a ⋅ b |a||b| = a x b x + a y b y + a z b z a 2 x + a 2 y + a 2 z b 2 x + b 2 y + b 2 z .(4.5.3.3) Properties of scalar product: 1. a ⋅ b = b ⋅ a (commutativity). 2. a ⋅ (b + c)=a ⋅ b + a ⋅ c (distributivity with respect to addition of vectors). This property holds for any number of summands. 3. If vectors a and b are collinear, then a ⋅ b = |a||b|. (The sign + is taken if the vectors a and b have the same sense, and the sign – is taken if the senses are opposite.) 4. (λa) ⋅ b = λ(a ⋅ b) (associativity with respect to a scalar factor). 5. a ⋅ a = |a| 2 . The scalar product a ⋅ a is denoted by a 2 (the scalar square of the vector a). 4.5. COORDINATES,VECTORS,CURVES, AND SURFACES IN SPACE 121 6. The length of a vector is expressed via the scalar product by |a| = √ a ⋅ a = √ a 2 . 7. Two nonzero vectors a and b are perpendicular if and only if ab = 0. 8. The scalar products of basis vectors are i ⋅ j = i ⋅ k = j ⋅ k = 0, i ⋅ i = j ⋅ j = k ⋅ k = 1. 9. If vectors are given by their coordinates, a =(a x , a y , a z )andb =(b x , b y , b z ), then a ⋅ b =(a x i + a y j + a z j)(b x i + b y j + b z j)=a x b x + a y b y + a z b z .(4.5.3.4) 10. The Cauchy–Schwarz inequality |a ⋅ b| ≤ |a||b|. 11. The Minkowski inequality |a + b| ≤ |a| + |b|. 4.5.3-2. Cross product of two vectors. The cross product of a vector a by a vector b is defined as the vector c (see Fig. 4.33) satisfying the following three conditions: 1. Its absolute value is equal to the area of the parallelogram spanned by the vectors a and b; i.e., |c| = |a × b| = |a||b| sin ϕ.(4.5.3.5) 2. It is perpendicular to the plane of the parallelogram; i.e., c⊥a and c⊥b. 3. The vectors a, b,andc form a right-handed trihedral; i.e., the vector c points to the side from which the sense of the shortest rotation from a to b is anticlockwise. Figure 4.33. Cross product of two vectors. Remark 1. The cross product of a vector a by a vector b is also denoted by c =[a, b]. Remark 2. If vectors a and b are collinear, then the parallelogram OADB is degenerate and should be assigned the zero area. Hence the cross product of collinear vectors is defined to be the zero vector whose direction is arbitrary. Properties of cross product: 1. a × b =–b × a (anticommutativity). 2. a × (b + c)=a × b + a × c (distributivity with respect to the addition of vectors). This property holds for any number of summands. . midpoint of a segment is equal to the half-sum of the corre- sponding coordinates of its endpoints. The angles α, β ,and between the segment −−−→ M 1 M 2 and the coordinate axes OX, OY , and OZ. e 33 z. (4.5.2.12) 4.5.2-4. Transformation of coordinates under translation and rotation of axes. Suppose that two rectangular Cartesian coordinate systems OXY Z and O X Y Z are given and the first system. Z and coordinates (ˆx,ˆy,ˆz) in the system O X Y Z. The transformation of rectangular Cartesian coordinates by the parallel translation and the rotation of axes is determined by the formulas x=e 11 ˆx