What is the projection of $< - 2, 4, 7 >$ $< -2, 4, 7>$ onto $< - 5, 9, - 5 >$ $< -5, 9, -5>$ ?

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Answer 1 · 2018-07-22T17:45:32

The projection is $= \frac{11}{131} \cdot < - 5, 9, - 5 >$ $=(11)/131*<-5,9,-5>$

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

$p r o j_{\to a} \to b = \frac{\to a . \to b}{{∣ ∣ \to a ∣ ∣}^{2}} \to a$ $proj_(veca)vecb=(veca.vecb)/(|veca|^2)veca$

Here,

$\to a = < - 5, 9, - 5 >$ $veca= <-5,9,-5>$

$\to b = < - 2, 4, 7 >$ $vecb = <-2,4,7>$

Therefore,

The dot product is

$\to a . \to b = < - 5, 9, - 5 > . < - 2, 4, 7 > = 10 + 36 - 35 = 11$ $veca.vecb=<-5,9,-5> . <-2,4,7> =10+36-35=11$

The modulus of $\to a$ $veca$ is

$∣ ∣ \to a ∣ ∣ = | < - 5, 9, - 5 > | = \sqrt{25 + 81 + 25} = \sqrt{131}$ $|veca| = |<-5,9,-5>| = sqrt(25+81+25)= sqrt131$

Therefore

$p r o j_{\to a} \to b = \frac{11}{131} \cdot < - 5, 9, - 5 >$ $proj_(veca)vecb=(11)/131*<-5,9,-5>$

What is the projection of < -2, 4, 7><−2,4,7>< -2, 4, 7> onto < -5, 9, -5><−5,9,−5>< -5, 9, -5>?