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 R3? y = 7 2x+z = 8 1 x2 + y2 = 9 z ≥ y2 q 2 2 2 The distance between two points (x1; y1; z1) and (x2; y2; z2) is (x2 − x1) + (y2 − y1) + (z2 − z1) . x + x y + y z + z The midpoint of the points (x ; y ; z ) and (x ; y ; z ) is 1 2 ; 1 2 ; 1 2 . 1 1 1 2 2 2 2 2 2 The equation of a sphere with center (h; k; l) and radius r is: (x − h)2 + (y − k)2 + (z − l)2 = r2 Find an equation of the sphere that has center (3; −2; 7) and touches the xz-plane. 2 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 x2 + y2 + z2 + 6x − 8z = 11 Describe mathematically the top half of a solid sphere of radius 4 centered at the origin. 3 11.2 Vectors and the Dot Product In 3 dimensions a vector has 3 components: a =< a1; a2; a3 >. where a1 is the x-component, a2 is the y-component, and a3 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 =< a1; a2; a3 > can be written as a = a1i + a2j + a3k. q 2 2 2 The magnitude (or length) of the vector a =< a1; a2; a3 > is jaj = a1 + a2 + a3. −−! Given the points A(x1; y1; z1) and B(x2; y2; z2), the vector from A to B is AB =< x2 − x1; y2 − y1; z2 − z1 >. a To find a unit vector (vector of length 1) in the direction of a vector a, compute . jaj 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 Two vectors are parallel if one vector is a scalar multiple of the other. D 4 E For example, a = h4; −3; 6i is parallel to b = − 3 ; 1; −2 since a = −3b. Given two vectors a =< a1; a2; a3 > and b =< b1; b2; b3 >, the dot product (or scalar product) of a and b, denoted a · b, can be found in either of the following ways: a · b = jajjbj cos θ where θ is the angle between a and b a · b = a1b1 + a2b2 + a3b3 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 jaj · b would not since jaj is not a vector. The · here does not mean multiplication. It means dot product. The first formula above rearranged gives us a formula for finding the cosine of the angle between two nonzero vectors. a · b cos θ = jajjbj 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 6 ABC. 5 Given two vectors a =< a1; a2; a3 > and b =< b1; b2; b3 >, The scalar projection of b onto a is given by a · b compab = jaj The vector projection of b onto a is given by a · b a a · b projab = = a jaj jaj jaj2 Given the vectors a = 10i − 2k and b =< 3; −4; 1 >, find the scalar and vector projections of b onto a. 6 11.3 The Cross Product A determinant for a 2 × 2 array of numbers (matrix): a b = ad − bc c d 2 −3 Example: = 5 −9 A determinant for a 3 × 3 array of numbers (matrix): a a a 1 2 3 b2 b3 b1 b3 b1 b2 b1 b2 b3 = a1 − a2 + a3 c2 c3 c1 c3 c1 c2 c1 c2 c3 2 3 1 Example: 0 2 1 = 5 3 0 The cross product of two vectors a =< a1; a2; a3 > and b =< b1; b2; b3 > is: i j k a × b = a1 a2 a3 b1 b2 b3 Example: Find a × b if a =< 3; 2; −1 > and b =< 4; 1; 1 >. The cross product of two vectors is a VECTOR! 7 Very Important Fact: The vector a × b is orthogonal to both a and b. Find a vector that is perpendicular to the plane containing the points A(1; 2; 3);B(−2; 1; −1), and C(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 6= b × a. What is true is a × b = −b × a. If θ is the angle between two vectors a and b, then ja × bj = jajjbj sin θ This above fact tells us the following: (1) Two nonzero vectors a and b are parallel if and only if ja × bj = 0. (2) The area of the parallelogram formed by the vectors a and b is ja × bj. Find the area of the triangle from the previous example with vertices A(1; 2; 3);B(−2; 1; −1), and C(1; −1; 1). 8 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 = ja · (b × c)j 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 =< a1; a2; a3 >; b =< b1; b2; b3 >, and c =< c1; c2; c3 >. a a a 1 2 3 b1 b2 b3 c1 c2 c3 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.