Let a⃗ = î + ĵ + 2k̂, b⃗ = 2î - 3ĵ + k̂ and c⃗ = î - ĵ + k̂ be three given vectors. Let v⃗ be a vector in the plane of a⃗ and b⃗ whose projection on c⃗ is 2/√3. If v⃗ . ĵ = 7, then v⃗ . ( î + k̂ ) is equal to :

Solution

Maths sir · 5 steps

A vector in the plane of a and b is xa + yb, so turn both conditions into two equations in x and y.

1. Write v. Any vector in the plane of a⃗ and b⃗ is v⃗ = xa⃗ + yb⃗. Adding component by component, v⃗ = (x + 2y)î + (x - 3y)ĵ + (2x + y)k̂.
2. Use the projection. The projection of v⃗ on c⃗ is (v⃗ · c⃗)/|c⃗|, and |c⃗| = √3. Setting this equal to 2/√3 gives v⃗ · c⃗ = 2.

   v⃗ · c⃗ = (x + 2y) - (x - 3y) + (2x + y) = 2x + 6y = 2 ⇒ x + 3y = 1

3. Use the j-component. v⃗ · ĵ = x - 3y = 7. Adding this to x + 3y = 1 gives 2x = 8, so x = 4 and y = -1.
4. Find v. v⃗ = 4a⃗ - b⃗ = 2î + 7ĵ + 7k̂. Check: v⃗ · c⃗ = 2 - 7 + 7 = 2, as required.
5. So. v⃗ · (î + k̂) = 2 + 7 = 9, which is option (d).

The idea to keep. Whenever a vector lies in the plane of two given vectors, write it as xa⃗ + yb⃗. Each condition then becomes one linear equation in x and y.

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