The vector projection of #vecb# onto #veca# is
#proj_(veca)vecb=(veca.vecb)/(||veca||)^2*veca#
#veca= <2,1,2>#
#vecb= <3,3,-1>#
The dot product is
#veca.vecb= <2,1,2> . <3, 3,-1> = (2)*(3)+(1)*(3)+(2)*(-1)#
#=6+3-2=7#
The modulus of #veca# is
#||veca|| = ||<2,1,2>|| = sqrt((2)^2+(1)^2+(2)^2)#
#= sqrt(4+1+4)=sqrt(9)=3#
Therefore,
#proj_(veca)vecb=(7)/(3)^2* <2,1,2>#
#=7/9<2,1,2>#
The vector projection of #veca# onto #vecb# is
#proj_(vecb)veca=(veca.vecb)/(||vecb||)^2*vecb#
The modulus of #vecb# is
#||vecb|| = ||<3,3,-1>|| = sqrt((3)^2+(3)^2+(-1)^2)#
#= sqrt(9+9+1)=sqrt(19)#
#proj_(vecb)veca=(7)/(sqrt19)^2* <3,3,-1>#
#=7/19<3,3,-1>#
