Let a⃗, b⃗ and c⃗ be three unit vectors such that
| a⃗ - b⃗ |² + | a⃗ - c⃗ |² = 8.
Then | a⃗ + 2b⃗ |² + | a⃗ + 2c⃗ |² is equal to ______.

Solution

Maths sir · 3 steps

Each |a⃗ - b⃗|² is at most 4, so a sum of 8 forces b⃗ and c⃗ to equal -a⃗.

1. Expand the given. For unit vectors, |a⃗ - b⃗|² = 1 + 1 - 2 a⃗ · b⃗ = 2 - 2 a⃗ · b⃗. The same holds for c⃗. So the condition becomes 4 - 2(a⃗ · b⃗ + a⃗ · c⃗) = 8, which gives a⃗ · b⃗ + a⃗ · c⃗ = -2.
2. Use the limit. The dot product of two unit vectors is cos θ, so it is never less than -1. Two such numbers can add to -2 only if both equal -1. So a⃗ · b⃗ = a⃗ · c⃗ = -1, meaning b⃗ = c⃗ = -a⃗.

   |a⃗ + 2b⃗|² = |a⃗|² + 4 a⃗ · b⃗ + 4|b⃗|² = 1 - 4 + 4 = 1

3. So. In the same way |a⃗ + 2c⃗|² = 1. The required sum is 1 + 1 = 2.

The common slip: 6. Writing the cross term of |a⃗ + 2b⃗|² as 2 a⃗ · b⃗ instead of 4 a⃗ · b⃗ gives 10 - 4 = 6, not 2.

The idea to keep. When a sum hits its largest possible value, every term must be at its own maximum. For unit vectors, a⃗ · b⃗ = -1 means they point in exactly opposite directions.

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