If the vectors a⃗ = λ î + μ ĵ + 4k̂, b⃗ = - 2î + 4ĵ - 2k̂ and c⃗ = 2î + 3ĵ + k̂ are coplanar and the projection of a⃗ on the vector b⃗ is √54 units, then the sum of all possible values of λ + μ is equal to :

Solution

Maths sir · 3 steps

Coplanarity gives one line in λ and μ, the projection gives another: solve them together for each sign.

1. Coplanar condition. Three vectors are coplanar when their scalar triple product is zero. Expanding the determinant gives 10λ - 2μ - 56 = 0, that is, 5λ - μ = 28.

   |λ, μ, 4; -2, 4, -2; 2, 3, 1| = 10λ - 2μ - 56 = 0

2. Projection condition. |b⃗| = √(4 + 16 + 4) = √24 and a⃗ · b⃗ = -2λ + 4μ - 8. A projection of length √54 means |(-2λ + 4μ - 8)/√24| = √54, so -2λ + 4μ - 8 = ±√1296 = ± 36.

   Sign	Second equation	Solve with μ = 5λ - 28	λ + μ
   +36	-λ + 2μ = 22	λ = 26/3, μ = 46/3	24
   -36	-λ + 2μ = -14	λ = 14/3, μ = -14/3	0

3. So. The possible values of λ + μ are 24 and 0, and their sum is 24 + 0 = 24, option (a).

The idea to keep. Coplanar means the scalar triple product is zero. When a projection is given as a length, allow both signs of the dot product, then solve each case.

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