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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 ThreeDimensional 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 3D to remember: 1. ax by cz d + + = represents a plane 2. x a = is a surface parallel to yzplane 3. y b = is a surface parallel to zxplane 4. z c = is a surface parallel to xyplane 5. y x = is a vertical plane that intersects xyplane 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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 Spring '08
 Firoz
 Calculus, Vectors

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