Let a⃗ = 4î + 3ĵ and b⃗ = 3î - 4ĵ + 5k̂. If c⃗ is a vector such that c⃗ .( a⃗ × b⃗ ) + 25 = 0,c⃗ .(î + ĵ + k̂) = 4, and projection of c⃗ on a⃗ is 1, then the projection of c⃗ on b⃗ equals :

Solution

Maths sir · 4 steps

Turn each of the three conditions into a linear equation in the components of c, then solve.

1. Cross product first. With a⃗ = (4, 3, 0) and b⃗ = (3, -4, 5), the cross product is a⃗ × b⃗ = 15î - 20ĵ - 25k̂. Let c⃗ = xî + yĵ + zk̂.

   15x - 20y - 25z = -25 ⇒ 3x - 4y - 5z = -5 ; x + y + z = 4 ; (4x + 3y)/5 = 1 ⇒ 4x + 3y = 5

2. Solve for c. Put z = 4 - x - y into the first equation to get 8x + y = 15, so y = 15 - 8x. Then 4x + 3(15 - 8x) = 5 gives x = 2, y = -1, z = 3. So c⃗ = 2î - ĵ + 3k̂.
3. Project on b. c⃗ · b⃗ = 6 + 4 + 15 = 25 and |b⃗| = √(9 + 16 + 25) = 5√2.
4. So. The projection of c⃗ on b⃗ is 25/5√2 = 5/√2, which is option (d).

The idea to keep. Three independent scalar conditions on an unknown vector give three linear equations. Find the vector, and any projection after that is just (c⃗ · b⃗)/|b⃗|.

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