Find the area of the parallelogram whose adjacent sides are determined by the vectors a⃗ = î - ĵ + 3k̂ and b⃗ = 2î - 7ĵ + k̂.
The area of the parallelogram is 15 square units.
Explanation
This question is from Exercise 10.4 and tests the concept that the area of a parallelogram with adjacent sides represented by vectors a⃗ and b⃗ is given by |a⃗ × b⃗|. The solution requires computing the cross product using determinant method and then finding its magnitude.
Solution Steps
Step 1: The area of a parallelogram with adjacent sides a⃗ and b⃗ is given by |a⃗ × b⃗|.
Step 2: Compute the cross product a⃗ × b⃗ = |î ĵ k̂; 1 -1 3; 2 -7 1|
Step 3: Expanding the determinant: î[(-1)(1) - (3)(-7)] - ĵ[(1)(1) - (3)(2)] + k̂[(1)(-7) - (-1)(2)]
Step 4: Simplifying: î(-1 + 21) - ĵ(1 - 6) + k̂(-7 + 2) = 20î + 5ĵ - 5k̂
Step 5: Find magnitude: |a⃗ × b⃗| = ²) = = = 15