If the vectors, p⃗ = ( a + 1 )î + aĵ + ak̂,
q⃗ = aî + ( a + 1 )ĵ + ak̂ and
r⃗ = aî + aĵ + ( a + 1 )k̂( a ∈ R )
are coplanar and 3( p⃗ .q⃗ )² - λ | r⃗ × q⃗ |² = 0, then the value of λ is ______.

Solution

Maths sir · 4 steps

Coplanar means the determinant is zero. That fixes a, and then Lagrange's identity does the rest.

1. Coplanar condition. Three vectors are coplanar when their scalar triple product is zero. Add all three columns into the first, take out 3a+1, and subtract row 1 from rows 2 and 3. The determinant becomes (3a+1) × 1, so 3a+1=0 and a=-1/3.
2. The three vectors. With a=-1/3: p⃗=1/3(2,-1,-1), q⃗=1/3(-1,2,-1), r⃗=1/3(-1,-1,2). Each has |·|²=6/9=2/3, and any two of them have dot product (-2-2+1)/9=-1/3.

   |r⃗×q⃗|² = |r⃗|²|q⃗|²-(r⃗·q⃗)² = 4/9-1/9=1/3

3. First term. p⃗·q⃗=-1/3, so 3(p⃗·q⃗)²=3 × 1/9=1/3.
4. So. 1/3-λ × 1/3=0, which gives λ= 1.

The idea to keep. For |u⃗×v⃗|², do not expand the cross product. Use |u⃗|²|v⃗|²-(u⃗·v⃗)², which needs only lengths and dot products.

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