Question

If with reference to the right handed system of mutually perpendicular unit vectors i, j and k, vector α = 3i - j, vector β = 2i + j - 3k, then express vector β in the form vector β = vector(β₁ + β₂), where vector β₁ is parallel to vector α and vector β₂ is perpendicular to vector α.

Answer 1

Let vector β₁ = λ vector α, λ is a scalar, i.e. vector β₁ = 3λi - λj.

Now, vector β₂ = vector(β - β₁) = (2 - 3λ)i + (1 + λ)j - 3k

Now, since vector β₂ is perpendicular vector α, we should have vector (α x β) = 0 i.e.,

3(2 - 3λ) - (1 + λ) = 0

⇒ 6 - 9λ - 1 - λ = 0 ⇒ 5 - 10λ = 0

⇒ 10λ = 5 ⇒ λ = 5/10 = 1/2

Therefore vector β₁ = 3/2i - 1/2j and vector β₂ = 1/2i + 3/2j - 3k