Scalar and Projection

Question

Find the scalar and vector projection of the vector b = <-1,1,4> onto the vector a = <2,2,-5>.

I got the scalar projection to be -20/sqrt(33)

but I couldn't figure out how to get the vector projection. If you could explain how you get your answer, that would be appreciated!

Answer 1

The vector projection is a vector with magnitude of the scalar projection and the direction of the vector you project onto. In this case we need to get the component of b in the direction of a and then multiply it by a unit vector in the a direction. The length of the component of vector b in the direction of the of a from geometry is

|b| cosθ (draw a generic diagram),

and using the dot product formula gives,

|b| cosθ = a ⋅b/|a|. (scalar projection)

This agrees with your answer. Great work! To transform it into the vector projection we need to multiply by a unit vector in the a direction.

a ⋅b/|a| (a/|a|) = ( a ⋅b)/|a|² a.

This is the equation for the vector projection

proj_a(b) = ( a ⋅b)/|a|² a,

and,

( a ⋅b)/|a|² a = -20/33 <2,2,-5> = <-40/33,-40/33,100/33>.

Of course as always double check the algebra and arithmetic.