Let c⃗ be the projection vector of b⃗=λ î+4 k̂, λ>0, on the vector a⃗=î+2 ĵ+2 k̂. If |a⃗+c⃗|=7, then the area of the parallelogram formed by the vectors b⃗ and c⃗ is _________.

Solution

Maths sir · 4 steps

The projection is a multiple of a⃗, so |a⃗+c⃗|=7 pins down that multiple.

1. Write the projection. a⃗·b⃗ = λ + 8 and |a⃗|² = 1+4+4 = 9. So c⃗ = (λ+8)/9 a⃗, a scalar multiple of a⃗.
2. Use the length. a⃗+c⃗ = (1+(λ+8)/9)a⃗ and |a⃗| = 3. So 3(1+(λ+8)/9) = 7, since the bracket is positive for λ>0. This gives (λ+8)/9 = 4/3, so λ = 4.
3. Set up the area. Now b⃗ = 4î+4k̂ and c⃗ = 4/3a⃗. The area is |b⃗×c⃗| = 4/3|b⃗×a⃗|.

   b⃗×a⃗ = |î, ĵ, k̂; 4, 0, 4; 1, 2, 2| = -8î-4ĵ+8k̂

4. So. |b⃗×a⃗| = √(64+16+64) = 12, and the area is 4/3× 12 = 16.

The common slip: 12. Finding |b⃗×a⃗| and forgetting the factor 4/3 from c⃗ = 4/3a⃗ gives 12, not 16.

The idea to keep. A projection vector on a⃗ is always (a⃗·b⃗)/|a⃗|²a⃗. Treat it as a scalar times a⃗, and its lengths and cross products follow easily.

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