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Math 210, Quiz 1 (9/7/04) (1) Find the scalar and vector projections of vector < −1, 2, 3 > onto vector < 3, 1, 0 >. Answer: Let a =< −1, 2, 3 > and b =< 3, 1, 0 >. We like to find the scalar and vector projections from a onto b. (The answer is different for projection from b onto a. This is a common mistake.) Scalar projection from a onto b is: |a| cos θ = √ a·b = −1/ 10 . |b| Vector projection from a onto b is: ( a·b b 1 b 1 ) = −√ = − < 3, 1, 0 > . |b| |b| 10 10 |b| (2) Given coordinates of P , Q, R be (0,1,2), (2,4,5), (-1,0,1), respectively. Find a vector that is orthogonal to the plane formed by P, Q, R. Answer: P~Q =< 2, 4, 5 > − < 0, 1, 2 >=< 2, 3, 3 > and P~R =< −1, 0, 1 > − < 0, 1, 2 >=< −1, −1, −1 >. Now, P~Q × P~R =< 0, −1, 1 > . This vector is orthogonal to the plane formed by P, Q, R. 1