What is the projection of (4 i + 4 j + 2 k)(4i+4j+2k) onto (- 5 i + 4 j - 5 k)(−5i+4j−5k)?

1 Answer
Narad T.
Dec 27, 2017

The projection is =-7/33 <-5,4,-5>=−733<−5,4,−5>

Explanation:

The vector projection of vecb→b onto veca→a

proj_(veca)vecb=(veca.vecb)/(||veca||)vecaproj→a→b=→a.→b∣∣∣∣→a∣∣∣∣→a

Here,

vecb= <4,4,2>→b=<4,4,2>

veca= <-5,4,-5>→a=<−5,4,−5>

The dot product is

veca.vecb= <4,4,2>. <-5,4,-5> =(4*-5)+(4*4)+(2*-5)= -20+16-10=-14→a.→b=<4,4,2>.<−5,4,−5>=(4⋅−5)+(4⋅4)+(2⋅−5)=−20+16−10=−14

The modulus of vecb→b is

||veca||=sqrt((-5)^2+(4)^2+(-5)^2)=sqrt(66)∣∣∣∣→a∣∣∣∣=√(−5)2+(4)2+(−5)2=√66

Therefore,

proj_(veca)vecb=(-14)/(66)*<-5,4,-5>proj→a→b=−1466⋅<−5,4,−5>

=-7/33<-5,4,-5>=−733<−5,4,−5>

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