What is the projection of $< - 6, 1, 5 >$ $<-6,1,5 >$ onto $< 1, 5, 9 >$ $<1,5,9 >$ ?

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What is the projection of $< - 6, 1, 5 >$ $<-6,1,5 >$ onto $< 1, 5, 9 >$ $<1,5,9 >$ ?

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Answer 1 · 2016-10-27T17:55:40

The projection is $⟨ \frac{11}{27}, \frac{55}{27}, \frac{99}{27} ⟩$ $〈11/27,55/27,99/27〉$

Explanation:

Let $\to b = ⟨ - 6, 1, 5 ⟩$ $vecb=〈-6,1,5〉$ and $\to a = ⟨ 1, 5, 9 ⟩$ $veca=〈1,5,9〉$

Then, the vector projection of $\to b$ $vecb$ onto $\to a$ $veca$ is given by

$\frac{\to a . \to b}{∣ ∣ \to a ∣ ∣ \to a ∣ ∣} \cdot \to a$ $(veca.vecb)/(∣veca∣∣veca∣)*veca$

$\to a . \to b$ $veca.vecb$ is the dot product
and $∣ ∣ \to a ∣ ∣$ $∣veca∣$ is the modulus of the vector

the dot product is $\to a . \to b = ⟨ 1, 5, 9 ⟩ . ⟨ - 6, 1, 5 ⟩ = - 6 + 5 + 45 = 44$ $veca.vecb=〈1,5,9〉.〈-6,1,5〉=-6+5+45=44$

$∣ ∣ \to a ∣ = \sqrt{1^{2} + 5^{2} + 9^{2}} = \sqrt{1 + 25 + 82} = \sqrt{108}$ $∣veca∣= sqrt(1^2+5^2+9^2)=sqrt(1+25+82)=sqrt108$

so the projection $= (\frac{44}{108}) ⟨ 1, 5, 9 ⟩ = (\frac{11}{27}) ⟨ 1, 5, 9 ⟩ = ⟨ \frac{11}{27}, \frac{55}{27}, \frac{99}{27} ⟩$ $=(44/108)〈1,5,9〉=(11/27)〈1,5,9〉=〈11/27,55/27,99/27〉$

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What is the projection of $< - 6, 1, 5 >$ $<-6,1,5 >$ onto $< 1, 5, 9 >$ $<1,5,9 >$ ?

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Explanation:

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