If P⃗=3î+√3ĵ+2k̂ and Q⃗=4î+√3ĵ+2.5k̂ then, the unit vector in the direction of P⃗× Q⃗ is 1/x(√3î+ĵ-2√3k̂). The value of x is _________.

Solution

Physics sir · 4 steps

Find the cross product, pull out the bracket the question gives you, then x is just that bracket's length.

1. Set up the determinant. Write P⃗×Q⃗ as a determinant with î, ĵ, k̂ in the top row, then (3, √3, 2) and (4, √3, 2.5) below it.

   P⃗×Q⃗ = |î, ĵ, k̂; 3, √3, 2; 4, √3, 2.5| = √3/2 î + 1/2 ĵ - √3 k̂

2. Expand each component. For î: 2.5√3 - 2√3 = √3/2. For ĵ: -(3 × 2.5 - 2 × 4) = -(7.5 - 8) = 1/2. For k̂: 3√3 - 4√3 = -√3. So P⃗×Q⃗ = 1/2(√3 î + ĵ - 2√3 k̂).
3. Find the length. A unit vector is the vector divided by its own magnitude, so the factor 1/2 drops out. The magnitude of the bracket is √((√3)² + 1² + (2√3)²) = √(3 + 1 + 12) = √16 = 4.
4. So. The unit vector is 1/4(√3 î + ĵ - 2√3 k̂), so x = 4.

The common slip: 16. Forgetting the square root in the magnitude gives 3 + 1 + 12 = 16 as x, not 4. The magnitude is √16 = 4.

The idea to keep. Any positive number in front of a vector cancels when you form its unit vector. Pull out common factors first, and the unit vector is the leftover vector divided by its own magnitude.

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