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

1 Answer
Narad T.
Jul 21, 2017

The vector projection is =-74/59<-1,3,-7>=−7459<−1,3,−7>
The scalar projection is =-61/sqrt59=−61√59

Explanation:

Let vecb= <6,-4,8>→b=<6,−4,8> and veca= <-1,3,-7>→a=<−1,3,−7>

The vector projection of vecb→b over veca→a is

=(veca.vecb)/(||veca||^2)*veca=→a.→b∣∣∣∣→a∣∣∣∣2⋅→a

The dot product is

veca.vecb=<6,-4,8> . <-1,3,-7> =(6*-1)+(-4*3)+(8*-7)→a.→b=<6,−4,8>.<−1,3,−7>=(6⋅−1)+(−4⋅3)+(8⋅−7)

=-6-12-56=-74=−6−12−56=−74

The modulus of veca→a is

||<-1,3,-7>|| =sqrt(1+9+49) = sqrt59||<−1,3,−7>||=√1+9+49=√59

Therefore,

The vector projection is

=-74/59<-1,3,-7>=−7459<−1,3,−7>

The scalar projection is

=(veca.vecb)/(||veca||)=-61/sqrt59=→a.→b∣∣∣∣→a∣∣∣∣=−61√59

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