Let O be the origin. Let OP⃗ = xî + yĵ - k̂ and OQ⃗ = - î + 2ĵ + 3xk̂, x, y∈R, x > 0, be such that | PQ⃗ | = √20 and the vector OP⃗ is perpendicular OQ⃗. If OR⃗ = 3î + zĵ - 7k̂, z∈R, is coplanar with OP⃗ and OQ⃗, then the value of x² + y² + z² is equal to :

Solution

Maths sir · 4 steps

Use the two conditions to fix x and y first, then make the box product zero to find z.

1. Perpendicular condition. OP⃗ · OQ⃗ = -x + 2y - 3x = 0, so 2y = 4x, which gives y = 2x.
2. Length of PQ. PQ⃗ = (-1-x)î + (2-y)ĵ + (3x+1)k̂. Put y = 2x into |PQ⃗|² = 20: (x+1)² + (2-2x)² + (3x+1)² = 14x² + 6 = 20. So x² = 1, and since x > 0, x = 1 and y = 2.

   |3, z, -7; 1, 2, -1; -1, 2, 3| = 3(6+2) - z(3-1) - 7(2+2) = 0

3. Coplanar condition. Three vectors from O are coplanar when their scalar triple product is zero. With OP⃗ = î + 2ĵ - k̂ and OQ⃗ = -î + 2ĵ + 3k̂, the determinant gives 24 - 2z - 28 = 0, so z = -2.
4. So. x² + y² + z² = 1 + 4 + 4 = 9, which is option (b).

The idea to keep. Coplanar vectors give a zero scalar triple product, and perpendicular vectors give a zero dot product. Use each condition to remove one unknown at a time.

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