272_c12

272_c12 - Mat 272 Calculus III Updated on 1/27/08 Dr. Firoz...

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Unformatted text preview: Mat 272 Calculus III Updated on 1/27/08 Dr. Firoz Chapter 12 Vectors and Geometry of Space Section 12.1 Three-Dimensional Coordinate System Suppose 1 1 1 1 ( , , ) P x y z and 2 2 2 2 ( , , ) P x y z are the given points. Find the distance D. The distance D = 2 2 2 1 2 1 2 1 2 ( ) ( ) ( ) x x y y z z- +- +- Important equations in 3-D to remember: 1. ax by cz d + + = represents a plane 2. x a = is a surface parallel to yz-plane 3. y b = is a surface parallel to zx-plane 4. z c = is a surface parallel to xy-plane 5. y x = is a vertical plane that intersects xy-plane in the line y x = 6. 2 2 2 2 ( ) ( ) ( ) x a y b z c d- +- +- = is a sphere center at ( , , ) a b c and radius d 7. 2 2 2 2 ( ) ( ) ( ) x a y b z c d- +- +- = , z c is a hemisphere center at ( , , ) a b c Examples: 1. Find the equation of a sphere center at (1,2, 1)- and radius 1. Solution: 2 2 2 ( 1) ( 2) ( 1) 1 x y z- +- + + = 2. Determine whether the points lie on the straight line a) (5,1,3), (7,9 1), (1, 15,11) A B C-- Solution: Check that 2 21, 6 21, 4 21 AB BC AC = = = and AB AC BC + = , The points are on a line. b) (0,3, 4), (1,2, 2), (3,0,1) K L C-- . Like in a) you can show the points are not on the same line. 3. Find the center and radius of the sphere given by 2 2 2 6 4 2 11 x y z x y z + +- +- = Solution: Complete the square as 2 2 2 2 ( 3) ( 2) ( 1) 5 x y z- + + +- = and then center is at (3, 2,1)- and radius 5. 4. Describe in words the region of 3 represented by the equations or inequalities a) 5 y = - b) 5 x = c) 4 x > d) y e) 0 6 z f) y z = g) 2 2 2 1 x y z + + > h) xyz = i) 2 2 2 2 3 x y z z + +- < j) 2 2 1 x y + = k) 2 2 9 x z + Mat 272 Calculus III Updated on 1/27/08 Dr. Firoz Section 12.2 Vectors Parallelogram law: If we place two vectors , u v a a so that they start at a same point, then u v + a a lies along the diagonal of the parallelogram with , u v a a vectors as sides. For two vectors , , and , , u a b c v x y z =< > =< > a a the vector represented and defined by , , a AB x a y b z c = =< --- > ccca a and , , a BA a x b y c z- = =< --- > ccca a The length or magnitude of a vector: 2 2 2 ( ) ( ) ( ) a x a y b z c =- +- +- a Examples: 1. Given 3, 1 , 5,3 a b =<- > =< > a a . Find , 2 , 3 a b a b a b +-- a a a a a a and , 2 , 3 a b a b a b +-- a a a a a a 2. Find a vector that has same direction as the vector 2,4,5 < - > and magnitude 6. Solution: find unit vector 1 2,4,5 3 5 u = < - > a , the vector we are looking for is 2 6 2,4,5 5 w u = = < - > a a Section 12.3 The Dot Product of Vectors For two vectors , , , , , u a b c v x y z =< > =< > a a the dot product is defined as u v ax by cz = + + a a If is the angle between the vectors , u v a a , then cos u v u v = a a a a Case 1....
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272_c12 - Mat 272 Calculus III Updated on 1/27/08 Dr. Firoz...

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