The following two vectors are adjacent sides of a parallelogram: $\vec{A} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{B} = 5\hat{i} - 4\hat{k}$. The area of the parallelogram is ______
Official Answer Key:
19.30 to 19.50
Step-by-Step Solution & Explanation
Area of a parallelogram spanned by two adjacent side vectors = |A⃗ × B⃗|.
With A⃗ = (2, 3, −1) and B⃗ = (5, 0, −4):
A⃗ × B⃗ = î[(3)(−4) − (−1)(0)] − ĵ[(2)(−4) − (−1)(5)] + k̂[(2)(0) − (3)(5)]
= î(−12 − 0) − ĵ(−8 + 5) + k̂(0 − 15)
= −12î + 3ĵ − 15k̂
|A⃗ × B⃗| = √[(−12)² + 3² + (−15)²] = √(144 + 9 + 225) = √378 = 19.44 square units
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.