What is the projection of $<6,-4,8 >$ onto $<-1,3,-7 >$ ?

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Answer 1 · 2017-07-21T06:03:43

The vector projection is $=-74/59<-1,3,-7>$
The scalar projection is $=-61/sqrt59$

Let $vecb= <6,-4,8>$ and $veca= <-1,3,-7>$

The vector projection of $vecb$ over $veca$ is

$=(veca.vecb)/(||veca||^2)*veca$

The dot product is

$veca.vecb=<6,-4,8> . <-1,3,-7> =(6*-1)+(-4*3)+(8*-7)$

$=-6-12-56=-74$

The modulus of $veca$ is

$||<-1,3,-7>|| =sqrt(1+9+49) = sqrt59$

Therefore,

The vector projection is

$=-74/59<-1,3,-7>$

The scalar projection is

$=(veca.vecb)/(||veca||)=-61/sqrt59$

What is the projection of <6,-4,8 ><6,-4,8 > onto <-1,3,-7 ><-1,3,-7 >?