What is the projection of $< - 2, 4, 2 >$ $<-2,4,2 >$ onto $< 1, 8, - 3 >$ $<1,8,-3 >$ ?

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Answer 1 · 2017-10-14T09:11:24

The projection is $= \frac{12}{37} < 1, 8, - 3 >$ $=12/37<1,8,-3>$

Let the vectors be

$\to u = < - 2, 4, 2 >$ $vecu=<-2,4,2>$

and

$\to v = < 1, 8, - 3 >$ $vecv=<1,8,-3>$

The projection of $\to u$ $vecu$ onto $\to v$ $vecv$ is

$p r o j_{\to v} \to u = \frac{\to u . \to v}{{∣ ∣ ∣ ∣ \to v ∣ ∣ ∣ ∣}^{2}} \to v$ $proj_(vecv)vecu=(vecu.vecv)/(||vecv||^2)vecv$

$= \frac{< - 2, 4, 2 > . < 1, 8, - 3 >}{{| | < 1, 8, - 3 > | |}^{2}} < 1, 8, - 3 >$ $=(<-2,4,2>.<1,8,-3>)/(||<1,8,-3>||^2)<1,8,-3>$

$= \frac{- 2 + 32 - 6}{1 + 64 + 9} < 1, 8, - 3 >$ $=(-2+32-6)/(1+64+9)<1,8,-3>$

$= \frac{24}{74} < 1, 8, - 3 >$ $=24/74<1,8,-3>$

$= \frac{12}{37} < 1, 8, - 3 >$ $=12/37<1,8,-3>$

What is the projection of <−2,4,2><-2,4,2 > onto <1,8,−3><1,8,-3 >?