What is the projection of $< 5 , -6, 0 >$ onto $< 3 , -7, 4 >$ ?

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Answer 1 · 2016-07-17T17:37:13

$(-81/sqrt74,-189/sqrt74,108/sqrt74)$ .

Let $vecx=(5,-6,0)$ and $vecy=(3,-7,4)$

Then the Projection Vector of $vecx$ on $vecy$ , denoted by, $Proj_(vecy)vecx$ , and is defined by,

$Proj_(vecy)vecx=(vecx.vecy)/(||vecy||^2)vecy$

$={(5,6,0).(3,-7,4)}/{sqrt(3^2+(-7)^2+4^2)}(3,-7,4)$

$={(15-42)/sqrt74}(3,-7,4)$

$=(-27/sqrt74)(3,-7,4)$

$=(-81/sqrt74,-189/sqrt74,108/sqrt74)$ .

What is the projection of < 5 , -6, 0 >< 5 , -6, 0 > onto < 3 , -7, 4 >< 3 , -7, 4 >?