272_c12

# 272_c12 - Mat 272 Calculus III Updated on Dr Firoz Chapter...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ℜ 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 θ is the angle between the vectors , u v a a , then cos u v u v θ ⋅ = a a a a Case 1....
View Full Document

{[ snackBarMessage ]}

### Page1 / 8

272_c12 - Mat 272 Calculus III Updated on Dr Firoz Chapter...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online