The Three Dimensional Coordinate System

The Three-Dimensional Coordinate System The Three Dimensional Coordinate System You can construct a three-dimensional coordinate system by passing a z-axis perpendicular to both the x- and y-axes at the origin. Figure 11.1 shows the positive portion of each coordinate axis. Precalculus 11.1 Figure 11.1 The Three-Dimensional Coordinate System The Three-Dimensional Coordinate System In this section, you will work exclusively with right-handed systems, as illustrated in Figure 11.2. In a right-handed system, Octants II, III, and IV are found by rotating counterclockwise around the positive z-axis. Octant V is vertically below Octant I. See Figure 11.3. Figure 11.2 Octant I Octant II Octant III Octant IV Octant V Figure 11.3 The Three-Dimensional Coordinate System Example 1 Octants VI, VII, and VIII are then found by rotating counterclockwise around the negative z-axis. See Figure 11.3. Plot each point in space. a) (-1,3,4) b) (2,5,0) c) (1,3,6) d) (2,-1,4) Octant VI Octant VII Octant VIII Figure 11.3 1

The Distance and Midpoint Formulas Example 2 Find the distance between (3,1,4) and (0,-1,2). The Distance and Midpoint Formulas Example 3 Find the midpoint of the line segment joining (2,-2,4) and (1,3,6). The Equation of a Sphere Example 4 Find the standard equation of the sphere with center (1,5,-2) and radius 4. Does the sphere intersect the xy-plane? 2

Example 5 Find the center and radius of the sphere given 2 2 2 by x y z 4x 2y 8z 10 0 Example 6 Sketch the xy-trace of the sphere given by 2 2 2 2 x 2 y 3 z 6 7 (xy-trace: the intersection of the surface of the sphere with the xy-plane) Vectors in Space Vectors in Space Using the unit vectors i = 1, 0, 0, j = 0, 1, 0, and k = 0, 0, 1 in the direction of the positive z-axis, the standard unit vector notation for v is v = v 1 i + v 2 j + v 3 k as shown in Figure 11.14. Precalculus 11.2 Figure 11.14 Vectors in Space Vectors in Space If v is represented by the directed line segment from P(p 1, p 2, p 3 ) to Q(q 1, q 2, q 3 ), as shown in Figure 11.15, the component form of v is produced by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point v = v 1, v 2, v 3 = q 1 p 1, q 2 p 2, q 3 p 3. Figure 11.15 3

Example 1 Find the component form and length of vector v having initial point (2,4,1) and terminal point (3,5,2). Then find a unit vector in the direction of v. Example 2 Find the dot product of the two vectors: u 1,3,5 v 0,4, 2 Vectors in Space Example 3 Find the angle between the two vectors: v 0,1,5 u 4, 1, 3 If the dot product of two nonzero vectors is zero, the angle between the vectors is 90 (recall that cos 90 = 0). Such vectors are called orthogonal. For instance, the standard unit vectors i, j, and k are orthogonal to each other. Parallel Vectors Recall from the definition of scalar multiplication that positive scalar multiples of a nonzero vector v have the same direction as v, whereas negative multiples have the direction opposite that of v. Parallel Vectors In Figure 11.18, the vectors u, v, and w are parallel because u = 2v and w = v. In general, two nonzero vectors u and v are parallel if there is some scalar c such that u = cv. Figure 11.18 4

Example 4 Vector w has initial point (3,0,-1) and terminal point (2,1,4). Which of the following vectors is parallel to w? a) u 5,0,1 b) v 6, 6, 30 Example 5 Determine whether the points P(1,-1,3), Q(0,4,-2), and R(6,13,-5) are collinear. Example 6 The initial point of v 6, 1,4 is P(0,2,-3). What is the terminal point of the vector? Example 7 Solving an Equilibrium Problem A weight of 480 pounds is supported by three ropes. As shown in Figure 11.20, the weight is located at S(0, 2, 1). The ropes are tied to the points P(2, 0, 0), Q(0, 4, 0), and R( 2, 0, 0). Find the force (or tension) on each rope. Figure 11.20 The Cross Product The Cross Product of Two Vectors Precalculus 11.3 5

The Cross Product The Cross Product It is important to note that this definition applies only to three-dimensional vectors. The cross product is not defined for two-dimensional vectors. A convenient way to calculate u v is to use the following determinant form with cofactor expansion. (This 3 3 determinant form is used simply to help remember the formula for the cross product it is technically not a determinant because the entries of the corresponding matrix are not all real numbers.) The Cross Product Note the minus sign in front of the j-component. Recall that each of the three 2 2 determinants can be evaluated by using the following pattern. Example 1 Given u 2i j 3k and v i 3 j 2k, find each cross product. a) u v b) v u c) v v The Cross Product Geometric Properties of the Cross Product 6

Example 2 Find a unit vector that is orthogonal to both u 2i 3 j 4k and v i 2 j. Example 3 Show that the quadrilateral with vertices A(3,5,0), B(1,2,4), C(1,3,4) and D(3,6,0) is a parallelogram. Then find the area of the parallelogram. Is the parallelogram a rectangle? The Triple Scalar Product The Triple Scalar Product If the vectors u, v, and w do not lie in the same plane, the triple scalar product u (v w) can be used to determine the volume of the parallelepiped (a polyhedron, all of whose faces are parallelograms) with u, v, and w as adjacent edges, as shown in Figure 11.24. Area of base = v w Volume of parallelepiped = u (v w) Figure 11.24 The Triple Scalar Product Example 4 Find the volume of the parallelepiped having u i 2 j k, v i 2 j 2k, and w 2i k as adjacent edges. 7

Plane Curves Parametric Equations From this set of equations you can determine that at time t = 0, the object is at the point (0, 0). Similarly, at time t = 1, the object is at the point, and so on, as shown in Figure 10.52. Precalculus 10.6 Curvilinear Motion: Two Variables for Position, One Variable for Time Figure 10.52 Plane Curves Sketching a Plane Curve When sketching a curve represented by a pair of parametric equations, you still plot points in the xyplane. Each set of coordinates (x, y) is determined from a value chosen for the parameter t. Plotting the resulting points in the order of increasing values of t traces the curve in a specific direction. This is called the orientation of the curve. Example 1 Sketch the curve represented by the parametric equations. x t 2 t 2 y 2t Eliminating the Parameter Example 1 uses simple point plotting to sketch the curve. This tedious process can sometimes be simplified by finding a rectangular equation (in x and y) that has the same graph. This process is called eliminating the parameter. x = t 2 4 t = 2y x = (2y) 2 4 x = 4y 2 4 y = 8

Example 2 Sketch the curve represented by the equations by eliminating the parameter and adjusting the domain of the resulting rectangular equation. 1 x t y 2t 2 Example 3 Sketch the curve represented by x 2cos and y 2sin, 0 2 by eliminating the parameter. Example 4 Find a set of parametric equations to represent the graph of y 4x 3 using the following parameters. a) t x b) t 2 x Lines & Planes in Space Precalculus 11.4 Lines in Space In the plane, slope is used to determine an equation of a line. In space, it is more convenient to use vectors to determine the equation of a line. In Figure 11.26, consider the line L through the point P(x 1, y 1, z 1 ) and parallel to the vector v = a, b, c. Direction vector for L Figure 11.26 Lines in Space The vector v is the direction vector for the line L, and a, b, and c are the direction numbers. One way of describing the line L is to say that it consists of all points Q(x, y, z) for which the vector is parallel to v. This means that is a scalar multiple of v, and you can write = tv, where t is a scalar. = x x 1, y y 1, z z 1 = at, bt, ct = tv 9

Lines in Space By equating corresponding components, you can obtain the parametric equations of a line in space. Example 1 Find parametric and symmetric equations of the line L that passes through the point (3,2,1) and is parallel to v 1,3,5. If the direction numbers a, b, and c are all nonzero, you can eliminate the parameter t to obtain the symmetric equations of a line. Symmetric equations Example 2 Find a set of parametric and symmetric equations of the line that passes through the points (1,3,-2) and (4,0,1). Planes in Space You have seen how an equation of a line in space can be obtained from a point on the line and a vector parallel to it. You will now see that an equation of a plane in space can be obtained from a point in the plane and a vector normal (perpendicular) to the plane. Consider the plane containing the point P(x 1, y 1, z 1 ) having a nonzero normal vector n = a, b, c, as shown in Figure 11.28. Figure 11.28 Planes in Space This plane consists of all points Q(x, y, z) for which vector is orthogonal to n. Using the dot product, you can write n = 0 a, b, c x x 1, y y 1, z z 1 = 0 a(x x 1 ) + b(y y 1 ) + c(z z 1 ) = 0. the is orthogonal to n. The third equation of the plane is said to be in standard form. Planes in Space Regrouping terms yields the general form of the equation of a plane in space ax + by + cz + d = 0. General form of equation of plane Given the general form of the equation of a plane, it is easy to find a normal vector to the plane. Use the coefficients of x, y, and z to write n = a, b, c. 10

Example 3 Find the general equation of the plane passing through the points (3,2,2), (1,5,0) and (1,-3,1). Planes in Space Two distinct planes in three-space either are parallel or intersect in a line. If they intersect, you can determine the angle (0 90 ) between them from the angle between their normal vectors, as shown in Figure 11.30. Figure 11.30 Planes in Space Specifically, if vectors n 1 and n 2 are normal to two intersecting planes, the angle between the normal vectors is equal to the angle between the two planes and is given by Angle between two planes Example 4 Find the angle between the two planes given by 2x y 3z 0 and x 2y z 0, and find parametric equations of their line of intersection. Consequently, two planes with normal vectors n 1 and n 2 are 1. perpendicular if n 1 n 2 = 0. 2. parallel if n 1 is a scalar multiple of n 2. Sketching Planes in Space In figure 11.32, this process is continued by finding the yz-trace and the xz-trace and then shading the triangular region lying in the first octant. Distance Between a Point and a Plane The distance D between a point Q and a plane is the length of the shortest line segment connecting Q to the plane, as shown in Figure 11.35. (a) xy-trace (z = 0): 3x + 2y = 12 (b) yz-trace (x = 0): 2y + 4z = 12 Figure 11.32 (c) xz-trace (y = 0): 3x + 4z = 12 Figure 11.35 11

Distance Between a Point and a Plane If P is any point in the plane, you can find this distance by projecting the vector onto the normal vector n. The length of this projection is the desired distance. Example 5 Find the distance between the point Q(2,3,-1) and the plane 2x 3y 4z 8. 12