Question

What is the projection of `<-6,1,5 >` onto `<1,5,9 >`?

Answer 1

The projection is `〈11/27,55/27,99/27〉`

Explanation:

Let `vecb=〈-6,1,5〉` and `veca=〈1,5,9〉`

Then, the vector projection of `vecb` onto `veca` is given by

`(veca.vecb)/(∣veca∣∣veca∣)*veca`

`veca.vecb` is the dot product
and `∣veca∣` is the modulus of the vector

the dot product is `veca.vecb=〈1,5,9〉.〈-6,1,5〉=-6+5+45=44`

`∣veca∣= sqrt(1^2+5^2+9^2)=sqrt(1+25+82)=sqrt108`

so the projection `=(44/108)〈1,5,9〉=(11/27)〈1,5,9〉=〈11/27,55/27,99/27〉`