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272t1sol

# 272t1sol - Mat 272 Test 1 Dr Firoz 1 Find the center and...

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Mat 272 Test 1 Dr. Firoz 1 1. Find the center and radius of the sphere 2 2 2 4 4 4 8 16 1 0 x y z x y + + - + - = . Solution: 2 2 2 2 2 2 4( 2 8 ) 1 0 ( 1) ( 4) 21/ 4 x y z x y x y z + + - + - = - + + + = The center is at (1, -4, 0) and radius 21/ 2 2. Given a vector 4,4,3 r =< > . Find a vector of magnitude 10 in the direction of the given vector. Solution: Unit vector along 4,4,3 41 u < > = , vector with magnitude 10 is 40,40,30 10 41 v u < > = = 3. Find the scalar and vector projection of the vector 5,0,5 a =< - > onto 5,2, 1 b =< - - > Solution: Scalar projection of a onto b is 20 30 b a b comp a b = = and vector projection of a onto b is 2 20 5,2, 1 30 b a b proj a b b = = < - - > Where 25 0 5 20 25 4 1 30 a b and b = + - = = + + = 4. Find orthogonal projection of 5, 6, 6 b =< - - - > onto 1, 4,8 a =< - > The orthogonal projection of b onto a is 29 5, 6, 6 1, 4,8 81 a a orth b b proj b = - =< - - - > + < - > 5. Find the area of the parallelogram with vertices P(0, 0, 0), Q(-5, 1, 3), R(-5, 0, 1) and S(-10, 1, 4). Solution: 5,1,3 , 5,0,1 PQ PR =< - > =< - > and area of the parallelogram is 126 PQ PR × = where 10 5 PQ PR i j k × = - +

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Mat 272 Test 1 Dr. Firoz 2 6. Find a vector equation for the line through the point P(5, -2, -1) and parallel to the vector 2,3,2 v =< - > Solution: ( ) 5 2 , 2 3 , 1 2 r t t t t =< - - + - + > 7. Find the point of intersection between the lines
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272t1sol - Mat 272 Test 1 Dr Firoz 1 Find the center and...

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