Question

What is the projection of `<8,2,-6 >` onto `<5,-1,7 >`?

Answer 1

The vector projection is `〈-4/15,4/75,-28/75〉`

Explanation:

The vector projection of
`vecb` onto `veca`
is given by
`=(veca.vecb)/(∣veca∣^2)veca`

Let our vectors be `veca=〈a_1,a_2,a_3〉` and `vecb=〈b_1,b_2,b_3〉`

Then the dot product is `veca.vecb=〈a_1,a_2,a_3〉.〈b_1,b_2,b_3〉=a_1b_1+a_2b_2+c_3c_3`

And `∣veca∣^2=〈a_1,a_2,a_3〉.〈a_1,a_2,a_3〉=a_1a_1+a_2a_2+a_3c_3`

`veca=〈5,-1,7〉` and `vecb=〈8,2,-6〉`

Then the dot product is
`veca.vecb=〈5,-1,7〉.〈8,2,-6〉=40-2-42=-4`

and `∣veca∣^2=〈5,-1,7〉〈5,-1,7〉=25+1+49=75`

So the projection is `-4/75〈5,-1,7〉=〈-4/15,4/75,-28/75〉`