11.1 Three-Dimensional Coordinate System

Transcription

1 11.1 Three-Dimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and z-axes is shown below. The three axes divide the region into 8 octants. z = 0 is the equation of the xy-plane. y = 0 is the equation of the xz-plane. x = 0 is the equation of the yz-plane. What is the projection of the point (2,6, 5) onto the xy-plane? the yz-plane? the xz-plane? What do the following represent in R 3? y = 7 2x+z = 8 1

2 x 2 +y 2 = 9 z y 2 The distance between two points (x 1,y 1,z 1 ) and (x 2,y 2,z 2 ) is (x 2 x 1 ) 2 +(y 2 y 1 ) 2 +(z 2 z 1 ) 2. ( ) x1 +x 2 y 1 +y 2 z 1 +z 2 The midpoint of the points (x 1,y 1,z 1 ) and (x 2,y 2,z 2 ) is,, The equation of a sphere with center (h,k,l) and radius r is: (x h) 2 +(y k) 2 +(z l) 2 = r 2 Find an equation of the sphere that has center (3, 2,7) and touches the xz-plane at a single point. 2

3 Find the equation of the sphere that has diameter passing through the points (1,4, 10) and ( 3,6, 2). What is the intersection of this sphere with the yz-plane? Find the center and radius of the sphere x 2 +y 2 +z 2 +6x 8z = 11 Describe mathematically the top half of a solid sphere of radius 4 centered at the origin. 3

4 11.2 Vectors and the Dot Product In 3 dimensions a vector has 3 components: a = a 1,a 2,a 3. where a 1 is the x-component, a 2 is the y-component, and a 3 is the z-component. An equivalent way of writing a vector is by using the standard unit basis vectors: i = 1,0,0, j = 0,1,0, and k = 0,0,1. The vector a = a 1,a 2,a 3 can be written as a = a 1 i+a 2 j+a 3 k. The magnitude (or length) of the vector a = a 1,a 2,a 3 is a = a 2 1 +a2 2 +a2 3. Given the points A(x 1,y 1,z 1 ) and B(x 2,y 2,z 2 ), the vector from A to B is AB = x2 x 1,y 2 y 1,z 2 z 1. To find a unit vector (vector of length 1) in the direction of a vector a, compute a a. Let a be the vector from the point P(2, 4, 7) to the point Q(1,3, 5) and let b = 4i+2j 6k. Fid a unit vector in the same direction as a. Find a vector of length 4 in the same direction as the vector a+2b. 4

5 Two vectors are parallel if one vector is a scalar multiple of the other. For example, a = 4, 3,6 is parallel to b = 4 3,1, 2 since a = 3b. Given two vectors a = a 1,a 2,a 3 and b = b 1,b 2,b 3, the dot product (or scalar product) of a and b, denoted a b, can be found in either of the following ways: a b = a b cosθ where θ is the angle between a and b a b = a 1 b 1 +a 2 b 2 +a 3 b 3 A dot product can only be performed on two vectors and the result is a scalar. Note: In the context of this section, a b would make sense, but a b would not since a is not a vector. The here does not mean multiplication. It means dot product. The first formula above rearranged gives a formula for finding the cosine of the angle between two vectors. cosθ = a b a b Two vectors a and b are orthogonal (or perpendicular), if a b = 0. For what values of x are the vectors x,3x,4 and x,4,5 orthogonal? A triangle has vertices A(0,3,9),B(1, 2,1), and C(3,1,2). Find ABC. 5

6 Given two vectors a = a 1,a 2,a 3 and b = b 1,b 2,b 3, The scalar projection of b onto a is given by compab = a b a The vector projection of b onto a is given by ( ) a b a projab = a a = a b a 2 a Given the vectors a = 10i 2k and b = 3, 4,1, find the scalar and vector projections of b onto a. 6

7 11.3 The Cross Product A determinant for a 2 2 array of numbers (matrix): a b c d = ad bc 2 3 Example: 5 9 = A determinant for a 3 3 array of numbers (matrix): a 1 a 2 a 3 b b 1 b 2 b 3 = a 2 b 3 1 c c 1 c 2 c 3 2 c 3 a b 1 b 3 2 c 1 c 3 +a b 1 b 2 3 c 1 c Example: = The cross product of two vectors a = a 1,a 2,a 3 and b = b 1,b 2,b 3 is: a b = i j k a 1 a 2 a 3 b 1 b 2 b 3 Example: Find a b if a = 3,2, 1 and b = 4,1,1. The cross product of two vectors is a VECTOR! 7

8 Very Important Fact: The vector a b is orthogonal to both a and b. FindavectorthatisperpendiculartotheplanecontainingthepointsA(1,2,3),B( 2,1, 1), andc(1, 1,1). The direction in which the cross product points can be determined by the right-hand rule. The right-hand rule helps us to see that a b b a. What is true is a b = b a. If θ is the angle between two vectors a and b, then a b = a b sinθ This above fact tells us the following: (1) Two nonzero vectors a and b are parallel if and only if a b = 0. (2) The area of the parallelogram formed by the vectors a and b is a b. Findtheareaofthetrianglefromthepreviousexamplewithvertices A(1,2,3),B( 2,1, 1), andc(1, 1,1). 8

9 The scalar triple product of the vectors a,b, and c is a (b c). The volume of the parallelipiped determined by the vectors a,b, and c is the absolute value of the scalar triple product: V = a (b c) When finding the scalar triple product, you can either first find b c and then dot with a, or you can find it all in one step by computing the determinant below where a =< a 1,a 2,a 3 >,b =< b 1,b 2,b 3 >, and c =< c 1,c 2,c 3 >. a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 Find the volume of the parallelipiped formed by the vectors a = 1,3,1, b = 4, 1,2 and c = 2,2,0. What does it mean, then, if the scalar triple product of three vectors is 0? Do the points P(3,0,1), Q( 1,2,5), R(5,1, 1) and S(0,4,2) all lie in the same plane? i.e. Are they coplanar? 9