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Given the vectors v and w, write v = v∥ + vp, where v∥ is parallel to w, and vp is perpendicular to w:

a) v = (2, 3, -7); w = (1,-2,-5)

b) v = (-3, 1, 2); w = (8,5,-3)

I found this in an old midterm and I'm not sure how to progress. I know that to find a perpendicular vector the dot product must be 0 and that a parallel vector a multiple of the vector. Any help is appreciated.

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up vote 2 down vote accepted

You already have written the hints:

$v_{||}=\lambda w$ for some number $\lambda$, and $ v-v_{||} = v_\perp\ ( \perp w)$, so we have $$(v-\lambda w)\cdot w = 0$$ Now find $\lambda$.

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