Vectors and the geometry of space

1 Vectors and the Geometry of Space 1.1 Three-Dimensional Coordinate Systems In order to represent points in space, we first choose a fixed point O (the origin) and three directed lines through that are perpendicular to each other, called the coordinate axes and labeled the x-axis, y-axis, and z-axis To determine the orientation of a three-dimensional coordinate system, in this text we use the right-handed rule as illustrated in the following figure Now if P is any point in space, let a be the (directed) distance from the yz-plane to P, let b be the distance from the xz-plane to P and let c be the distance from the xy-plane to P We represent the point by the ordered triple (a, b, c) of real numbers and we call it the coordinates of P Distance Formula The distance d between two points ( x1 , y1 , z1 ) and ( x2 , y2 , z2 ) is d  ( x2  x1 )  ( y2  y1 )  ( z2  z1 ) Example Find the distance between the points (2, -1, 3) and (1, 0, -2) Example Find the midpoint of the line segment joining the points ( x1 , y1 , z1 ) and ( x2 , y2 , z2 ) Example Find an equation of a sphere with radius r and center ( x0 , y0 , z0 ) (the standard equation of a sphere) 1.2 Vectors The term vector is used by scientists to indicate a quantity that involves both magnitude and direction, such as force, velocity, and acceleration We denote a vector by printing a letter in boldface (v) or by putting an arrow above the letter ⃗⃗⃗⃗⃗⃗ has initial point (𝑣⃗) A vector is often represented by a directed line segment, for instance 𝑃𝑄 ⃗⃗⃗⃗⃗⃗ || P and terminal point Q, and its length (or magnitude) is denoted by ||𝑃𝑄 Vectors having the same length and the same direction (even though in a different position) are equivalent (or equal) and we write u = v The zero vector, denoted by 0, has length It is the only vector with no specific direction Components of a vector If the initial point of a vector is the origin of a rectangular coordinate system, then its terminal point has coordinates of the form (a1, a2) or (a1, a2, a3), depending on whether the system is two- or three-dimensional These coordinates are called the components of the vector Example: We can see in the following figure, all the geometric vectors are the representations of the algebraic vector 𝒂 = 〈3,2〉: In three dimensions, given the points A( x1, y1, z1 ) and B( x2 , y2 , z2 ) , the vector a with representation ⃗⃗⃗⃗⃗⃗ 𝐴𝐵 is Properties of vectors in components form • Equality of vectors: Let u  u1 , u2 , u3 and v  v1 , v2 , v3 be vectors in space u = v if and only if u1 = v1, u2 = v2 and u3 = v3 • The magnitude or length of the vector v  v1 , v2 , v3 is the length of any of its representations: • The vector sum of u  u1 , u2 , u3 and v  v1 , v2 , v3 is the vector: u + v  u1  v1 , u2  v2 , u3  v3 • To multiple a vector u  u1 , u2 , u3 by a scalar c: cu  cu1 , cu2 , cu3 and length of a scalar multiple: ||cu|| = |c| ||u|| If a is nonzero, the vector a = a ||a|| ||a|| has length and the same direction as a, called unit vector in the direction of a • The negative of u is the vector: u= - u = (-1)u  u1 , u2 , u3 • The difference of u and v is: u - v = u + (-v)  u1  v1 , u2  v2 , u3  v3 Properties of Vector Operations Let u, v and w be vectors in the plane, and let c and d be scalars The standard basis vectors have length and point in the directions of the positive x-, y-, and z-axes If a  a1 , a2 , a3 , then we can write Thus any vector in space can be expressed in terms of i, j, and k Similarly, in two dimensions, we can write Where i = 〈1,0〉 and j = 〈0,1〉 Vectors have many applications in physics and engineering One example is force A vector can be used to represent force, because force has both magnitude and direction If two or more forces are acting on an object, then the resultant force on the object is the vector sum of the vector forces Example Two tugboats are pushing an ocean liner, as shown in the figure Each boat is exerting a force of 400 pounds What is the resultant force on the ocean liner? Solution Using the figure, you can represent the forces exerted by the first and second tugboats as The resultant force on the ocean liner is So, the resultant force on the ocean liner is approximately 752 pounds in the direction of the positive x-axis 1.3 The dot product So far, we have two operations with vectors—vector addition and multiplication by a scalar—each of which yields another vector In this section, you will study a third vector operation, the dot product This product yields a scalar, rather than a vector The dot product of u  u1 , u2 , u3 and v  v1 , v2 , v3 is This operation has some properties: Let u, v, and w be vectors in the plane or in space and let c be a scalar The angle between two nonzero vectors u and v is the angle 𝜃, ≤ 𝜃 ≤ 𝜋, between their respective standard position vectors u.v = ||u|| ||v|| cos 𝜃 or, if u and v are nonzero, The vectors u and v are orthogonal when u.v = Direction Angles and Direction Cosines The direction angles of a nonzero vector v are the angles α, β, and γ (in the interval [0,π]) that v makes with the positive x-, y-, and z-axes, or with three unit vectors i, j, and k The cosines of these direction angles, cos α, cos β, and cos γ, are called the direction cosines of the vector v cos 𝛼 = 𝐯∙𝐢 𝑣1 = ‖𝐯‖‖𝐢‖ ‖𝐯‖ cos 𝛽 = 𝐯∙𝐣 𝑣2 = ‖𝐯‖‖𝐣‖ ‖𝐯‖ cos 𝛾 = 𝐯∙𝐤 𝑣3 = ‖𝐯‖‖𝐤‖ ‖𝐯‖ Similarly, Consequently, any nonzero vector v in space has the normalized form and because v/||v|| is a unit vector, it follows that Example: Find the direction cosines and angles for the vector v = 2i + 3j + 4k Projections The vector projection of vector b onto vector a is denoted by projab The scalar projection of b onto a (also called the component of b along a) is defined to be the signed magnitude of the vector projection: compab Using the dot product, we get: comp𝐚 𝐛 = |𝐛| cos 𝜃 = 𝐚∙𝐛 𝐚 = ∙𝐛 |𝐚| |𝐚| Notice that the vector projection is the scalar projection times the unit vector in the direction of a Example: Find the scalar projection and vector projection of b = 〈1,1,2〉 onto a = 〈−2,3,1〉 Work The work W done by the constant force F acting along the line of motion of an object is given by When the constant force F is not directed along the line of motion, the work W done by the force is Thus, the work done by a constant force as its point of application moves along the vector is one of the following Example: To close a sliding door, a person pulls on a rope with a constant force of 50 pounds at a constant angle of 60o as shown in the following figure Find the work done in moving the door 12 feet to its closed position Solution Using a projection, you can calculate the work as follows 1.4 Cross Product Many applications in physics, engineering, and geometry involve finding a vector inspace that is orthogonal to two given vectors In this section, you will study a product that will yield such a vector Definition If u  u1 , u2 , u3 and v  v1 , v2 , v3 , then th cross product of u and v is the vector u × v  u2v3  u3v2 , u3v1  u1v3 , u1v2  u2v1 A convenient way to calculate is to use the determinant form with cofactor expansion shown below Note the minus sign in front of the j-component Example: Algebraic Properties of the Cross Product Geometric Properties of the Cross Product The Triple Product For vectors u, v, and w in space, the triple scalar product is: Geometric Property of the Triple Scalar Product The volume V of a parallelepiped with vectors u, v, and w as adjacent edges is V = |u ∙ (v × w)| 1.5 Lines and Planes The parametric equations of a line parallel to the vector 𝐯 = 〈𝑎, 𝑏, 𝑐 〉 and passing through the point P(x0, y0, z0) is: The symmetric equations of the line can be obtained by eliminating the parameter t, if the direction numbers a, b, and c are all nonzero, The standard equation of a plane containing the point (x0, y0, z0) and having normal vector 𝐧 = 〈𝑎, 𝑏, 𝑐 〉 is The linear equation (or general form of the equation) of a plane can be obtained by regrouping terms, ax + by + cz + d = Example: Find the general equation of the plane containing the points (2, 1, 1), (0, 4, 1), (-2, 1, 4) Example: Find the angle between the two planes x – 2y + z = and 2x + 3y – 2z = Then find parametric equations of their line of intersection Distances Between Points, Planes, and Lines Thus the distance between the point (x1, y1, z1) and the plane having the linear equation ax + by + cz + d = is Example: Find the distance between the parallel planes 10x + 2y - 2z = and 5x + y – z = 1.6 Cylinders and Quadratic Surfaces Cylinders A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve Example: z = x2 is a parabolic cylinder x2 + y2 = Quadric Surfaces z2 + y2 = A quadric surface is the graph of a second-degree equation in three variables x, y, and z The most general such equation is There are six basic types of quadric surfaces:  ... be vectors in the plane, and let c and d be scalars The standard basis vectors have length and point in the directions of the positive x-, y-, and z-axes If a  a1 , a2 , a3 , then we can write... Note the minus sign in front of the j-component Example: Algebraic Properties of the Cross Product Geometric Properties of the Cross Product The Triple Product For vectors u, v, and w in space, the. .. Equality of vectors: Let u  u1 , u2 , u3 and v  v1 , v2 , v3 be vectors in space u = v if and only if u1 = v1, u2 = v2 and u3 = v3 • The magnitude or length of the vector v  v1 , v2 , v3 is the