Quiz 3 Solutions

Quiz 3 Solutions - -y = z Now to make c a unit vector we...

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Quiz 3 Solutions Let a = < 1 , 1 , 0 > , and b = < 0 , 1 , 1 > in the following two problems 1. Calculate: (1) 3 a - 2 b (2) a.b (3) comp a b (4)proj a b Sol : (1) 3 a - 2 b =3 < 1 , 1 , 0 > - 2 < 0 , 1 , 1 > = < 3 , 1 , - 2 > (2) a.b = < 1 , 1 , 0 >< 0 , 1 , 1 > =(1)(0) + (1)(1) + (0)(1) = 1 (3) comp a b = ab | a | = 1 2 = 2 2 , since | a | = 1 2 + 1 2 + 0 = 2 (4)proj a b =comp a b . a | a | = 2 2 . < 1 , 1 , 0 > 2 = < 1 2 , 1 2 , 0 > 2. Find a unit vector, say c , which is orthogonal to both a and b . Sol : Assume c = < x, y, z > . We have: < x, y, z > . < 1 , 1 , 0 > = x + y = 0, since c is orthogonal to a Similarly, we have < x, y, z > . < 0 , 1 , 1 > = y + z = 0, since c is orthogonal to b In summary, we have
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Unformatted text preview: -y = z . Now, to make c a unit vector, we need x 2 + y 2 + z 2 = 1, then 3 x 2 = 1, after plugging in the identity we just arrived. then , x = √ 3 3 , y =-√ 3 3 , and z = √ 3 3 , or x =-√ 3 3 , y = √ 3 3 , and z =-√ 3 3 , i.e c = < √ 3 3 ,-√ 3 3 , √ 3 3 > , or c = <-√ 3 3 , √ 3 3 ,-√ 3 3 >...
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