Question

If a⃗, b⃗, c⃗ and d⃗ are four non-zero vectors such that a⃗ × b⃗ = c⃗ × d⃗ and a⃗ × c⃗ = 4b⃗ × d⃗, then show that (a⃗ − 2d⃗) is parallel to (2b⃗ − c⃗) where a⃗ ≠ 2d⃗, c⃗ ≠ 2b⃗.

The vectors a = 2î – 4ĵ + λk̂ and b = 3î – 6ĵ + k̂ are collinear if value of λ is :

Assertion (A) : A line through the points (4, 7, 8) and (2, 3, 4) is parallel to a line through the points (–1, –2, 1) and (1, 2, 5). Reason (R) : Lines r = a₁ + λb₁ and r = a₂ + μb₂ are parallel if b₁ . b₂ = 0.

Two vectors →a = a₁î + a₂ĵ + a₃k̂ and →b = b₁î + b₂ĵ + b₃k̂ are collinear if