Question

Find the vector projection of B onto A if A = 5i + 11j – 2k,B = 4i + 7k​

Answer 1

`(3)/(13) i+(11)/(130) j-(6)/(65) k\\`

Explanation:

Given A = 5i + 11j – 2k and B = 4i + 7k​, the vector projection of B unto a is expressed as
`proj_ab = (b.a)/(||a||^2) * a`

b.a = (5i + 11j – 2k)*( 4i + 0j + 7k)

note that i.i = j.j = k.k =1

b.a = 5(4)+11(0)-2(7)

b.a = 20-14

b.a = 6

||a|| = √5²+11²+(-2)²

||a|| = √25+121+4

||a|| = √130

square both sides

||a||² = (√130)

||a||² = 130

`proj_ab = (6)/(130) * (5i+11j-2k)\\\\proj_ab = (30)/(130) i+(11)/(130) j-(12)/(130) k\\\\proj_ab = (3)/(13) i+(11)/(130) j-(6)/(65) k\\\\`

Hence the projection of b unto a is expressed as
`(3)/(13) i+(11)/(130) j-(6)/(65) k\\`