Vectors and the Geometry of Space

Question

Let a=(−5,0,1) and b=(5,−8,3) be vectors.

(A) Find the scalar projection of b onto a.

Scalar Projection:

(B) Decompose the vector b into a component parallel to a and a component orthogonal to a.

Parallel component: ( , , )

Orthogonal Component: ( , , )

Answer 1

(A) proj_ab = (a·b)/IaI = (- 25 + 3)/√26 = - 22/√26

(B) Parallel component: - 22√26·<- 5, 0, 1> = <110/√26, 0, - 22/√26>.

Orthogonal Component: <5, - 8, 3> - <110/√26, 0, - 22/√26> = <5 - 110/√26, - 8, 3 + 22/√26>