If a⃗ = 2î + ĵ + 3k̂, b⃗ = 3î + 3ĵ + k̂ and c⃗ = c₁î + c₂ĵ + c₃k̂ are coplanar vectors and a⃗ . c⃗ = 5, b⃗ ⊥ c⃗, then 122(c₁ + c₂ + c₃) is equal to ___________.

Solution

Maths sir · 4 steps

Write c⃗ as xa⃗ + yb⃗, and the two dot conditions hand you x and y.

1. Use coplanarity. a⃗ and b⃗ are not parallel, so any vector in their plane is c⃗ = xa⃗ + yb⃗. Now we only need x and y, not three separate components.
2. Dot products needed. a⃗ · a⃗ = 4 + 1 + 9 = 14, b⃗ · b⃗ = 9 + 9 + 1 = 19 and a⃗ · b⃗ = 6 + 3 + 3 = 12.
3. Two equations. a⃗ · c⃗ = 14x + 12y = 5 and b⃗ · c⃗ = 12x + 19y = 0. The second gives y = -12x/19. Putting this in the first gives 122x/19 = 5, so x = 95/122 and y = -60/122.

   c₁ + c₂ + c₃ = x(2+1+3) + y(3+3+1) = (6 × 95 - 7 × 60)/122 = 150/122

4. So. 122(c₁ + c₂ + c₃) = 150, so the answer is 150.

The idea to keep. Coplanar with two given vectors means c⃗ = xa⃗ + yb⃗. Every dot condition on c⃗ then becomes a linear equation in x and y.

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