Let a⃗ = î + α ĵ + 3k̂ and b⃗ = 3î - α ĵ + k̂. If the area of the parallelogram whose adjacent sides are represented by the vectors a⃗ and b⃗ is 8√3 square units, then a⃗ . b⃗ is equal to __________.

Solution

Maths sir · 4 steps

The area hands you α², and α² is all the dot product needs.

1. Cross product. Expand the determinant with rows (1, α, 3) and (3, -α, 1). This gives a⃗ × b⃗ = 4α î + 8 ĵ - 4α k̂.
2. Use the area. The parallelogram's area is |a⃗ × b⃗| = 8√3. Square both sides so the root goes away.

   16α² + 64 + 16α² = 192 ⇒ 32α² = 128 ⇒ α² = 4

3. Dot product. a⃗ · b⃗ = (1)(3) + (α)(-α) + (3)(1) = 6 - α². Only α² is needed, not the sign of α.
4. So. a⃗ · b⃗ = 6 - 4 = 2.

The common slip: -16. Using the triangle formula, 1/2|a⃗ × b⃗| = 8√3, gives α² = 22 and a dot product of -16, not 2.

The idea to keep. Parallelogram area is the full |a⃗ × b⃗|; the triangle is half of it. Square the magnitude to get a clean equation in the unknown.

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