Question

If a and b are non-zero and non-parallel vectors such that λa+ub is parallel to ha+tb, where λ ,u, h, b €R and h is not 0, b is not 0, show that λ/h = u/t​

Answer 1

Explanation:

Since λa + ub is parallel to ha + tb, we can write:

λa + ub = k(ha + tb)

where k is some constant. We can simplify this equation by expanding both sides:

λa + ub = kha + ktb

Rearranging the terms, we get:

λa - kha = ktb - ub

Factoring out a, we get:

a(λ - kh) = b(kt - u)

Since a and b are non-parallel, they are linearly independent, which means that a and b are not multiples of each other. This also means that the only way for the left side of the equation to be equal to 0 is if λ - kh = 0, or λ = kh. Similarly, the only way for the right side of the equation to be equal to 0 is if kt - u = 0, or u/t = k/h.

Therefore, we have shown that λ/h = u/t.