Let p⃗ = 2î + 3ĵ + k̂ and q⃗ = î + 2ĵ + k̂ be two vectors. If a vector r⃗ = (α î + β ĵ + γ k̂) is perpendicular to each of the vectors ((p⃗ + q⃗ ) and (p⃗ - q⃗ ), and | r⃗ | = √3, then | α | + | β | + | γ | is equal to _______________.

Solution

Maths sir · 3 steps

A vector perpendicular to two given vectors points along their cross product, so find that and rescale it.

1. Find the two vectors. p⃗ + q⃗ = 3î + 5ĵ + 2k̂ and p⃗ - q⃗ = î + ĵ. Since r⃗ is perpendicular to both, it must lie along their cross product.

   (p⃗ + q⃗) × (p⃗ - q⃗) = |î, ĵ, k̂; 3, 5, 2; 1, 1, 0| = -2î + 2ĵ - 2k̂

2. Fix the length. This cross product has length √(4 + 4 + 4) = 2√3, but r⃗ needs length √3. Halve it: r⃗ = ±(-î + ĵ - k̂).
3. So. Either sign gives |α| = |β| = |γ| = 1, so |α| + |β| + |γ| = 3.

The common slip: 6. Taking the cross product itself as r⃗ without scaling it to length √3 gives 2 + 2 + 2 = 6, not 3.

The idea to keep. Perpendicular to two vectors means parallel to their cross product. Find the direction first, then scale it to the required length. The sign ambiguity disappears once absolute values are taken.

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