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

1 Answer
Oct 10, 2016

"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>