What is the projection of #< 5 ,- 3, 9 ># onto #< 8 , 8 , -4 >#?

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Answer 1 · 2017-12-15T16:09:59

The projection is #= -5/3 <8,8, -4>#

The vector projection of #vecv# onto #vecu# is

#proj_uv= (vec u. vecv ) /(||vecu||)^2*vecu#

Here,

#vecv= < 5,-3,9> # and

#vecu = <8,8,-4>#

The dot product is

#vecu.vecv =<5,-3,9> . <8,8,-4> #

#=(5*8)+(-3*8)+(9*-4) =40-24-36=-20#

The magnitude of #vecu# is

#=||<8,8,-4 >|| = sqrt((8)^2+(8)^2+(-4)^2)#

#=sqrt(64+64+16)=sqrt144=12#

The vector projection is

#proj_uv=-20/12* <8,8,-4> =-5/3 <8,8, -4>#