What is the projection of $(- i + j + k)$ $(-i + j + k)$ onto $(i - j + k)$ $( i - j + k)$ ?

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What is the projection of $(- i + j + k)$ $(-i + j + k)$ onto $(i - j + k)$ $( i - j + k)$ ?

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Answer 1 · 2016-05-30T07:53:52

The projection of a vector a onto vector b is given by

$p r o j_{a} b = \frac{a \cdot b}{{| a |}^{2}} \cdot a$ $proj_a b=(a*b)/absa^2 *a$

Hence

The dot product of $a = (- 1, 1, 1)$ $a=(-1,1,1)$ and $b = (1, - 1, 1)$ $b=(1,-1,1)$ is

$a \cdot b = - 1 - 1 + 1 = - 1$ $a*b=-1-1+1=-1$

The magnitude of a is $| a | = \sqrt{- 1^{2} + 1^{2} + 1^{2}} = \sqrt{3}$ $absa=sqrt(-1^2+1^2+1^2)=sqrt3$

Hence the projection is

$p r o j_{a} b = - \frac{1}{3} \cdot (- 1, 1, 1) = (- \frac{1}{3}, \frac{1}{3}, \frac{1}{3}) = \frac{1}{3} \cdot (- i + j + k)$ $proj_a b=-1/3*(-1,1,1)=(-1/3,1/3,1/3)=1/3*(-i+j+k)$

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What is the projection of $(- i + j + k)$ $(-i + j + k)$ onto $(i - j + k)$ $( i - j + k)$ ?

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