Let the vectors a⃗, b⃗, c⃗ be such that
| a⃗ | = 2, | b⃗ | = 4 and | c⃗ | = 4. If the projection of
b⃗ on a⃗ is equal to the projection of c⃗ on a⃗
and b⃗ is perpendicular to c⃗, then the value of
| a⃗ + b⃗ - c⃗ | is ___________.

Solution

Maths sir · 4 steps

Square the magnitude and expand. The two given conditions make every cross term vanish.

1. Read the conditions. The projection of b⃗ on a⃗ is (a⃗ · b⃗)/|a⃗|, and the same holds for c⃗. Equal projections give a⃗ · b⃗ = a⃗ · c⃗. Since b⃗ ⊥ c⃗, we also have b⃗ · c⃗ = 0.

   |a⃗ + b⃗ - c⃗|² = |a⃗|² + |b⃗|² + |c⃗|² + 2 a⃗ · b⃗ - 2 a⃗ · c⃗ - 2 b⃗ · c⃗

2. Cross terms cancel. 2 a⃗ · b⃗ - 2 a⃗ · c⃗ = 0 because the two dot products are equal, and 2 b⃗ · c⃗ = 0. Only the squared lengths remain.
3. Put in numbers. |a⃗ + b⃗ - c⃗|² = 2² + 4² + 4² = 4 + 16 + 16 = 36.
4. So. |a⃗ + b⃗ - c⃗| = √36 = 6.

The common slip: 36. Stopping at the squared magnitude gives 36. Take the square root at the end to get the length.

The idea to keep. To find the length of a sum of vectors, square it and expand with dot products. Conditions like perpendicular or equal projections are there to kill the cross terms.

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