Let a⃗ be a vector which is perpendicular to the vector 3î + 1/2ĵ + 2k̂. If a⃗ × ( 2î + k̂ ) = 2î - 13ĵ - 4k̂, then the projection of the vector a⃗ on the vector 2î + 2ĵ + k̂ is :

Solution

Maths sir · 4 steps

Write a⃗ as (x, y, z), expand the cross product, and the two conditions pin down all three components.

1. Expand the cross product. Let a⃗ = xî + yĵ + zk̂. Crossing it with 2î + k̂ gives the determinant below.

   a⃗ × (2î + k̂) = |î, ĵ, k̂; x, y, z; 2, 0, 1| = y î + (2z - x) ĵ - 2y k̂

2. Match components. Comparing with 2î - 13ĵ - 4k̂ gives y = 2, 2z - x = -13 and -2y = -4. The last one only confirms y = 2, so you still need one more equation.
3. Use perpendicularity. The dot product with 3î + 1/2ĵ + 2k̂ is zero, so 3x + 1 + 2z = 0. Putting x = 2z + 13 into it gives 8z + 40 = 0, so z = -5 and x = 3. That makes a⃗ = 3î + 2ĵ - 5k̂.
4. So. With b⃗ = 2î + 2ĵ + k̂ and |b⃗| = 3, the projection is (a⃗ · b⃗)/|b⃗| = (6 + 4 - 5)/3 = 5/3, which is option (c).

The idea to keep. A cross product equation gives only two independent equations, because one component is always repeated. You need one more condition, such as a dot product, to fix the vector completely.

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