Question

What is the projection of `< 5 ,- 3, 2 >` onto `< -1 , -2 , -4 >`?

Answer 1

The vector projection is `= <1/3,2/3,4/3>`

Explanation:

The vector projection of `vecb` onto `veca` is

`proj_(veca)vecb=(veca.vecb)/(||veca||)^2*veca`

`veca= <-1,-2,-4>`

`vecb= <5,-3,2>`

The dot product is

`veca.vecb= <-1,-2,-4> . <5, -3,2> = (-1)*(5)+(-2)*(-3)+(-4)*(2)`

`=-5+6-8=-7`

The modulus of `veca` is

`||veca|| = ||<-1,-2,-4>|| = sqrt((-1)^2+(-2)^2+(-4)^2)`

`= sqrt(1+4+16)=sqrt(21)`

Therefore,

`proj_(veca)vecb=(-7)/(sqrt(21))^2* <-1,-2,-4>`

`=-1/3<-1,-2,-4>`

`= <1/3,2/3,4/3>`