# 13 - If s is scalar s = s Proving iff statement two vectors...

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If s is scalar, s = s Proving iff statement: two vectors v+w are orthogonal iff v w = 0. 1. Assume v+w are orthogonal. Then angle between them, , is 2 . 2. So v w = v w cos 2 = 0. ...3. Assume v w = 0. So v w cos = 0. Since v+w are nonzero, v 0 and w 0. So cos = 0 w/ 0 . = 2 ie. the vectors are orthogonal. v x w 2 = v 2 w 2 v w 2 . 13.5 Equations of Lines and Planes Two lines are parallel their direction numbers are scalar multiples. Two lines intersect their x, y, and z-values are identical at some t-value. Ex. L 1 : x 1 1 2 = y 1 1 1 = z 1 2 4 = t and L 2 : x 2 2 4 = y 2 3 = z 2 1 2 1 = t . Solve for x , y , z . Set up a system of equations with x 1 = x 2 , y 1 = y 2 , z 1 = z 2 and solve. If you get an answer, they are , they’re skew. EXAMPLE 5 Find an equation of the plane that passes through the points P(1, 3, 2), Q(3,1,6), and R(5, 2, 0). First, solve for vectors PQ and PR , using P as the arbitrary starting point. No normal vector given, but PQ x PR gives one. EXAMPLE 6 Find the point at which the line with parametric equations

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13 - If s is scalar s = s Proving iff statement two vectors...

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