Let three vectors a⃗ ,b⃗ and c⃗ be such that c⃗ is coplanar
with a⃗ and b⃗, a⃗ .c⃗ = 7 and b⃗ is perpendicular to c⃗, where
a⃗ = - î + ĵ + k̂ and b⃗ = 2î + k̂ , then the
value of 2| a⃗ + b⃗ + c⃗ |² is _____.

Solution

Maths sir · 4 steps

Coplanar means c⃗ = xa⃗ + yb⃗; the two dot conditions give you x and y.

1. Coplanar form. Since c⃗ lies in the plane of a⃗ and b⃗, write c⃗ = xa⃗ + yb⃗. You need a⃗ · b⃗ = -2 + 0 + 1 = -1, |a⃗|² = 3 and |b⃗|² = 5.
2. Use both conditions. b⃗ · c⃗ = 0 gives x(-1) + 5y = 0, so x = 5y. Then a⃗ · c⃗ = 7 gives 3x - y = 7, so 14y = 7. This makes y = 1/2 and x = 5/2.
3. Find c. c⃗ = 5/2(-î + ĵ + k̂) + 1/2(2î + k̂) = -3/2î + 5/2ĵ + 3k̂. Check: b⃗ · c⃗ = -3 + 3 = 0 and a⃗ · c⃗ = 3/2 + 5/2 + 3 = 7.

   a⃗ + b⃗ + c⃗ = -1/2î + 7/2ĵ + 5k̂

4. So. |a⃗ + b⃗ + c⃗|² = 1/4 + 49/4 + 25 = 75/2, and twice that is 75.

The common slip: 37.5. Stopping at |a⃗ + b⃗ + c⃗|² gives 37.5. The question asks for twice this value.

The idea to keep. Whenever a vector is coplanar with two given vectors, write it as xa⃗ + yb⃗. Each dot-product condition then becomes one linear equation in x and y.

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