Decompose vector

Question

let u=i+6j and v=-5+5j. Decompose u into two vectors u_1 and u_2 where u_1 is parrallel to v and u_2 is orthogonal to v. Write answer in the form ai+bj

u_1=

u_2=

Answer 1

OK. For each vector, the slope is equal to y-component (factor of j) over x-component (factor of i).

v has a slope of -1. u₁ is parallel, so it must have a slope of -1. Its x- and y-components are negative of one another.

u₁ = a i - a j

u₂ is orthogonal to v, so it will have a slope that is the negative reciprocal of u₁. It's slope is 1. Its x- and y-components are equal to each other, but not necessarily to those of u₁.

u₂ = b i + b j

If you add u1 and u2 you get u. So the components of u1 and u2 come out like this:

u1 + u2 = (a + b) i + (-a + b) j = 1 + 6 j

So a + b = 1 and -a + b = 6. Solve this pair and get

2a = -5 or a = -2.5

2b = 7 or b = 3.5

This leaves you with u₁ = -2.5 i + 2.5 j and u₂ = 3.5 i + 3.5 j

If you add them together you get u. So this is your answer.