What is the projection of (-i + j + k)(−i+j+k) onto ( i - j + k)(i−j+k)?

1 Answer
Konstantinos Michailidis
May 30, 2016

The projection of a vector a onto vector b is given by

proj_a b=(a*b)/absa^2 *aprojab=a⋅b|a|2⋅a

Hence

The dot product of a=(-1,1,1)a=(−1,1,1) and b=(1,-1,1)b=(1,−1,1) is

a*b=-1-1+1=-1a⋅b=−1−1+1=−1

The magnitude of a is absa=sqrt(-1^2+1^2+1^2)=sqrt3|a|=√−12+12+12=√3

Hence the projection is

proj_a b=-1/3*(-1,1,1)=(-1/3,1/3,1/3)=1/3*(-i+j+k)projab=−13⋅(−1,1,1)=(−13,13,13)=13⋅(−i+j+k)

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