What is the projection of $< - 5, 3, 7 >$ $<-5,3,7 >$ onto $< 0, 8, - 2 >$ $<0,8,-2 >$ ?

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Answer 1 · 2016-11-26T06:18:41

The projection is $= ⟨ 0, \frac{20}{17}, - \frac{5}{17} ⟩$ $=〈0,20/17,-5/17〉$

Let $\to a = ⟨ 0, 8, - 2 ⟩$ $veca=〈0,8,-2〉$

and $\to b = ⟨ - 5, 3, 7 ⟩$ $vecb=〈-5,3,7〉$

The projection of $\to b$ $vecb$ onto $\to a$ $veca$ is

$= \frac{\to a . \to b}{∥ ∥ \to a {∥ ∥}^{2}} \to a$ $=(veca.vecb)/(∥veca∥^2)veca$

Let's calculate the dot product

$veca.vecb=〈0,8,-2〉.〈-5,3,7〉=0*-5+8*3*-2*7=0+24-14=10$

Then, we calculate the modulus of $veca$

$∥veca∥=∥〈0,8,-2〉∥=sqrt(0+64+4)=sqrt68$

The ptojection is $=10/68〈0,8,-2〉=5/34〈0,8,-2〉$

$=〈0,40/34,-10/34〉$

$=〈0,20/17,-5/17〉$

What is the projection of <-5,3,7 ><−5,3,7><-5,3,7 > onto <0,8,-2 ><0,8,−2><0,8,-2 >?