What is the projection of #< 5 , -6, 0 ># onto #< 3 , -7, 4 >#?

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Answer 1 · 2016-07-17T17:37:13

#(-81/sqrt74,-189/sqrt74,108/sqrt74)#.

Let #vecx=(5,-6,0)# and #vecy=(3,-7,4)#

Then the Projection Vector of #vecx# on #vecy#, denoted by, #Proj_(vecy)vecx#, and is defined by,

#Proj_(vecy)vecx=(vecx.vecy)/(||vecy||^2)vecy#

#={(5,6,0).(3,-7,4)}/{sqrt(3^2+(-7)^2+4^2)}(3,-7,4)#

#={(15-42)/sqrt74}(3,-7,4)#

#=(-27/sqrt74)(3,-7,4)#

#=(-81/sqrt74,-189/sqrt74,108/sqrt74)#.