What is the projection of #<4 , -6, 7 ># onto #< -1 , 9,-2 >#?

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Answer 1 · 2017-07-22T14:32:20

The vector projection is #=-72/86<-1,9,-2>#
The scalar projection is #=-72/sqrt86#

Let #vecb= <4,-6,7># and #veca= <-1,9,-2>#

The vector projection of #vecb# over #veca# is

#=(veca.vecb)/(||veca||^2)*veca#

The dot product is

#veca.vecb=<4,-6,7> . <-1,9,-2> =(4*-1)+(-6*9)+(7*-2)#

#=-4-54-14=-72#

The modulus of #veca# is

#||<-1,9,-2>|| =sqrt(1+81+4) = sqrt86#

Therefore,

The vector projection is

#=-72/86<-1,9,-2>#

The scalar projection is

#=(veca.vecb)/(||veca||)=-72/sqrt86#