What is the projection of $< 6, 0, 3 >$ $<6,0,3 >$ onto $< 2, 5, - 7 >$ $<2,5,-7 >$ ?

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Answer 1 · 2016-10-21T20:09:09

The vector projection is $⟨ - \frac{9}{39}, - \frac{45}{78}, \frac{63}{78} ⟩$ $〈-9/39,-45/78,63/78〉$

Let $\to b = ⟨ 6, 0, 3 ⟩$ $vecb=〈6,0,3〉$

and $\to a = ⟨ 2, 5, - 7 ⟩$ $veca=〈2,5,-7〉$

The dot product is $\to a . \to b = (6 \cdot 2 + 5 \cdot 0 + 3 \cdot - 7) = 12 - 21 = - 9$ $veca.vecb=(6*2+5*0+3*-7)=12-21=-9$

$∣ ∣ \to a ∣^{2} = (4 + 25 + 49) = 78$ $∣veca∣^2= (4+25+49)=78$

So the projection is
$(\to a . \to b) \frac{\to a}{∣ ∣ \to a {∣ ∣}^{2}} = - \frac{9}{78} ⟨ 2, 5, - 7 ⟩$ $(veca.vecb)veca / (∣veca∣^2)=-9/78 〈2,5,-7〉$

What is the projection of <6,0,3><6,0,3 > onto <2,5,−7><2,5,-7 >?