Let a⃗ , b⃗ and c⃗ be three unit vectors such that
a⃗ + b⃗ + c⃗ = 0⃗. If λ = a⃗ .b⃗ + b⃗.c⃗ + c⃗ .a⃗ and
d⃗ = a⃗ × b⃗ + b⃗ × c⃗ + c⃗ × a⃗ , then the ordered pair, ( λ ,d⃗ ) is equal to :

Solution

Maths sir · 4 steps

Square the sum to get λ, then replace c by −(a + b) and every cross product collapses to a × b.

1. Square the sum. Take the dot product of a⃗ + b⃗ + c⃗ = 0⃗ with itself. Each unit vector gives |a⃗|² = |b⃗|² = |c⃗|² = 1, and every mixed term appears twice.

   0 = 1 + 1 + 1 + 2(a⃗·b⃗ + b⃗·c⃗ + c⃗·a⃗) = 3 + 2λ

2. Find λ. So 2λ = -3, which gives λ = -3/2. It must be negative, because the three vectors cancel each other out.
3. Rewrite each cross product. Put c⃗ = -(a⃗ + b⃗). Then b⃗ × c⃗ = -b⃗ × a⃗ = a⃗ × b⃗, and c⃗ × a⃗ = -b⃗ × a⃗ = a⃗ × b⃗, because a⃗ × a⃗ = b⃗ × b⃗ = 0⃗.
4. So. d⃗ = a⃗ × b⃗ + a⃗ × b⃗ + a⃗ × b⃗ = 3 a⃗ × b⃗. The pair is (-3/2, 3 a⃗ × b⃗), which is option (c).

The idea to keep. When vectors add up to zero, square the sum to get the dot products. For the cross products, write one vector in terms of the others, and remember that swapping the order flips the sign.

Answer from the NTA final answer key. Solution by PrepWiser.