Let O be the origin and the position vector of the point P be - î - 2ĵ + 3k̂. If the position vectors of the points A, B and C are - 2î + ĵ - 3k̂,2î + 4ĵ - 2k̂ and - 4î + 2ĵ - k̂ respectively, then the projection of the vector OP⃗ on a vector perpendicular to the vectors AB⃗ and AC⃗ is :

Solution

Maths sir · 4 steps

Cross AB with AC to get the perpendicular vector, then project OP onto it.

1. Two edge vectors. AB⃗ = B - A = 4î + 3ĵ + k̂ and AC⃗ = C - A = -2î + ĵ + 2k̂.

   n⃗ = AB⃗ × AC⃗ = |î, ĵ, k̂; 4, 3, 1; -2, 1, 2| = 5î - 10ĵ + 10k̂

2. Length of the normal. |n⃗| = √(25 + 100 + 100) = √225 = 15. This vector is perpendicular to both AB⃗ and AC⃗.
3. Projection. OP⃗ · n⃗ = (-1)(5) + (-2)(-10) + (3)(10) = -5 + 20 + 30 = 45. The projection is (OP⃗ · n⃗)/|n⃗| = 45/15.
4. So. The projection of OP⃗ is 3, which is option (b).

The idea to keep. Perpendicular to two vectors means their cross product. Projection of a⃗ on b⃗ is (a⃗ · b⃗)/|b⃗|, so always divide by the length of b⃗.

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