Assignment3 Practice

Assignment3 Practice - ~ y = 0. 7. Find a scalar equation...

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Math 136 Practice Problems # 3 1. Let ~u = 3 2 - 2 , ~v = 1 3 - 1 , and ~w = 3 1 2 . a) Find a unit vector in the direction of ~u . b) Determine the angle between ~v and ~w . 2. Determine all values of k for which 1 2 1 and k 2 k 4 are orthogonal. 3. a) Let ~u = - 1 1 2 and ~v = 2 3 1 . Calculate proj ~v ~u and perp ~v ~u . b) Let ~u = 2 - 1 3 4 and ~v = 1 0 0 1 . Calculuate proj ~v ~u and perp ~v ~u . 4. Find the scalar equation of the plane passing through P ( - 3 , - 3 , 1) with normal vector ~n = - 1 4 7 . 5. Let ~a R 3 . Prove that proj ~a ( perp ~a ( ~x ) ) = ~ 0 for any ~x R 3 . 6. Let ~x,~ y R n . Prove that k ~x + ~ y k 2 = k ~x k 2 + k ~ y k 2 if and only if ~x ·
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Unformatted text preview: ~ y = 0. 7. Find a scalar equation of the plane with vector equation ~x = 4-2-5 + s -1 3-2 + t -2 1 2 . 8. Let ~v = 1 1 1 , ~u = 1-2 1 and ~w = 1 . Consider the plane with vector equation ~x = s~v + t~u. a) Determine a scalar equation of the plane. b) Determine the projection of ~w onto the plane by using your result form part a). c) Evaluate proj ~v ~w + proj ~u ~w . 1...
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