Let a⃗=î+ĵ+k̂, b⃗=3 î+2 ĵ-k̂, c⃗=λ ĵ+μ k̂ and d be a unit vector such that a⃗ × d=b⃗ × d and c⃗ · d=1. If c⃗ is perpendicular to a⃗, then |3 λ d+μ c⃗|² is equal to________

Solution

Maths sir · 4 steps

Bring both cross products to one side, and d must lie along a⃗-b⃗.

1. Find the direction. From a⃗ × d = b⃗ × d we get (a⃗-b⃗) × d = 0⃗, so d is parallel to a⃗-b⃗ = -2î-ĵ+2k̂. Its length is √(4+1+4) = 3, so d = ±1/3(-2î-ĵ+2k̂).
2. Use both conditions. c⃗ ⊥ a⃗ gives λ + μ = 0, so μ = -λ. Then c⃗ · d = ±1/3(-λ + 2μ) = ±1/3(-3λ) = ∓λ, and this must equal 1.
3. Take the plus sign. With d = 1/3(-2î-ĵ+2k̂) we get λ = -1, μ = 1, so c⃗ = -ĵ+k̂. The minus sign gives λ = 1, μ = -1, and the vector below comes out exactly the same.

   3λd + μc⃗ = (2î+ĵ-2k̂) + (-ĵ+k̂) = 2î - k̂

4. So. |3λd + μc⃗|² = 2² + 0² + (-1)² = 5.

The idea to keep. When two cross products with the same vector are equal, subtract them. A zero cross product means the two vectors are parallel, and that fixes the direction up to a sign.

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