Let a be a unit vector perpendicular to the vectors b⃗ = î - 2ĵ + 3k̂ and c⃗ = 2î + 3ĵ - k̂ , and a makes an angle of cos⁻¹ ( -1/3 ) with the vector î + ĵ + k̂ . If a makes an angle of π/3 with the vector î + αĵ + k̂ , then the value of a is:

Solution

Maths sir · 5 steps

Cross b⃗ with c⃗ for the direction, let the angle pick the sign, then solve for α.

1. Direction of a. A vector perpendicular to both b⃗ and c⃗ lies along b⃗ × c⃗ = ((-2)(-1) - (3)(3))î - ((1)(-1) - (3)(2))ĵ + ((1)(3) - (-2)(2))k̂ = -7î + 7ĵ + 7k̂. So a = ±1/√3(-î + ĵ + k̂).
2. Fix the sign. With the plus sign, a · (î + ĵ + k̂) = 1/√3, so the cosine of the angle is 1/√3 ÷ √3 = 1/3. The question needs -1/3, so take the minus sign: a = 1/√3(î - ĵ - k̂).
3. Use the angle. Take v⃗ = î + αĵ + k̂. Then a · v⃗ = (1 - α - 1)/√3 = -α/√3 and |v⃗| = √(2 + α²). Since cos π/3 = 1/2, the dot product gives the equation below.

   -α/√3 = 1/2√(2 + α²)

4. Solve with care. The right side is positive, so α must be negative. Squaring gives α²/3 = (2 + α²)/4, so 4α² = 6 + 3α² and α² = 6.
5. So. α = -√6, which is option (c).

The idea to keep. Perpendicular to two vectors means along their cross product, up to a plus or minus sign. Use any extra angle condition to fix that sign, and check signs again before you square an equation.

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