Question

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

Answer 1

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)`