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

1 Answer
Jul 12, 2016

The projection is (-15/sqrt46 , 45/sqrt46 , -90/sqrt46).

Explanation:

Given two vectors vecV and vecW, the projection of vecV onto vecW is given by:

(vecV * vecW)vecW/||vecW||

The inner product gives the component of VecV in the direction of VecW, and the fraction performs the same function as multiplying this magnitude by a unit vector in the direction of vecW.

So plugging in given values:
vecV = (3,4,-1)
vecW = (-1,3,-6)
||vecW|| = sqrt(1 + 9 + 36) = sqrt(46)

vecV*vecW = (3*-1) + (4*3) + (-1*-6)
= -3 + 12 + 6 = 15

So plugging everything in, the projection is:
(15*(-1,3,-6))/sqrt(46) = (-15/sqrt46 , 45/sqrt46 , -90/sqrt46)