Let A(2,3,5) and C(-3,4,-2) be opposite vertices of a parallelogram A B C D. If the diagonal BD⃗=î+2 ĵ+3 k̂, then the area of the parallelogram is equal to :

Solution

Maths sir · 4 steps

Both diagonals are known, so the area is half the size of their cross product. No need to find B or D.

1. Area from diagonals. For a parallelogram with diagonal vectors d₁⃗ and d₂⃗, the area is 1/2|d₁⃗ × d₂⃗|. Here d₁⃗ = AC⃗ and d₂⃗ = BD⃗.
2. First diagonal. AC⃗ = C - A = (-3-2)î + (4-3)ĵ + (-2-5)k̂ = -5î + ĵ - 7k̂.

   AC⃗ × BD⃗ = |î, ĵ, k̂; -5, 1, -7; 1, 2, 3| = 17î + 8ĵ - 11k̂

3. Find the magnitude. The î part is 3 + 14 = 17, the ĵ part is -(-15 + 7) = 8 and the k̂ part is -10 - 1 = -11. The magnitude is √(289 + 64 + 121) = √474.
4. So. The area is 1/2√474, which is option (d).

The idea to keep. If the diagonals are given, the area is half their cross product. If two adjacent sides are given, the area is the full cross product. Check which pair you have before you use the formula.

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