What is the projection of $<8,2,-6 >$ onto $<5,-1,7 >$ ?

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Answer 1 · 2016-10-25T06:48:49

The vector projection is $〈-4/15,4/75,-28/75〉$

The vector projection of
$vecb$ onto $veca$
is given by
$=(veca.vecb)/(∣veca∣^2)veca$

Let our vectors be $veca=〈a_1,a_2,a_3〉$ and $vecb=〈b_1,b_2,b_3〉$

Then the dot product is $veca.vecb=〈a_1,a_2,a_3〉.〈b_1,b_2,b_3〉=a_1b_1+a_2b_2+c_3c_3$

And $∣veca∣^2=〈a_1,a_2,a_3〉.〈a_1,a_2,a_3〉=a_1a_1+a_2a_2+a_3c_3$

$veca=〈5,-1,7〉$ and $vecb=〈8,2,-6〉$

Then the dot product is
$veca.vecb=〈5,-1,7〉.〈8,2,-6〉=40-2-42=-4$

and $∣veca∣^2=〈5,-1,7〉〈5,-1,7〉=25+1+49=75$

So the projection is $-4/75〈5,-1,7〉=〈-4/15,4/75,-28/75〉$

What is the projection of <8,2,-6 ><8,2,-6 > onto <5,-1,7 ><5,-1,7 >?