How to Calculate Area of a Parallelogram with Vectors
Finding the area of a parallelogram spanned by u→ and v→ is the same as calculating the length of the u→ ×v→-vector.
If you know the angle between the two vectors, you can use this formula:
If you have the vectors on vector coordinate form, you use this formula:
Find the area of the parallelogram that is spanned by u→ = (1, 3,−2) and v→ = (−3, 2, 4).
You start by finding the cross product:
If you have two vectors a→ and b→, where |a→| = 5, |b→| = 7, and the angle between them is 30°, you can find the area of the parallelogram they span by inserting your information into the formula: |a→ ×b→| = |a→| ⋅ |b→| ⋅ sin α = 5 ⋅ 7 ⋅ sin 30° = 5 ⋅ 7 ⋅1 2 = 35 2 = 17.5.
The area of the parallelogram is equal to 17.5.