Question

What is the projection of `< 5 ,- 3, 9 >` onto `< 5, -2 , -4 >`?

Answer 1

`"proj"_v u = color(green)(< -5/9,2/9,4/9>)`

Explanation:

In general for vectors `vecu` and `vecv`
the projection of `vecu` unto `vecv` is given by
`color(white)("XXX")"proj_v u = ( (vecu * vecv)/(abs(abs(vecv))^2)) * vecv`

`color(white)("XXXXXXXXXXXX")`[If you need a derivation of this equation,
`color(white)("XXXXXXXXXXXXX")`ask as a separate question.]

For the given case
`color(white)("XXX")vecu= < 5,-3,9>` and
`color(white)("XXX")vecv = <5,-2,-4>`

`vecu * vecv = (5 * 5) + ((-3) * (-2)) + (9 * (-4))`
`color(white)("XXXX") = 25 +6 -36`
`color(white)("XXXX") = -5`

`abs(abs(vecv))^2 = 5^2+(-2)^2+(-4)^2`
`color(white)("XXX")=25+4+16`
`color(white)("XXX")=45`

`((vecu * vecv)/(abs(abs(vecv))^2)) = (-5)/(45)`
`color(white)("XXXXXX")=-1/9`

`"proj"_v u = (-1/9)vecv`
`color(white)("XXX")=(-1/9) * < 5, -2, -4>`
`color(white)("XXX")=<-5/9, 2/9, 4/9>`