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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.

Two vectors and the area of the parallelogram spanned by the vectors

If you know the angle between the two vectors, you can use this formula:

Formula

Area of a Parallelogram with a Known Angle

|u ×v| = |u| |v| sin α,α = (u,v)

If you have the vectors on vector coordinate form, you use this formula:

Formula

Area of a Parallelogram on Vector Coordinate Form

|u ×v| = | (x1, y1, z1 ) × ( x 2, y2, z2 ) | = |(y1z2 y2z1,z1x2 z2x1, x1y2 x2y1)|

|u ×v| = | (x1, y1, z1 ) × ( x 2, y2, z2 ) | = | (y1z2 y2z1,z1x2 z2x1,x1y2 x2y1) |

Example 1

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:

u ×v = ( 1 1, 3, 2 ) × ( 3, 2, 1 4 ) = (3 4 2 (2), (2) (3) 1 4, 1 2 (3) 3 ) = (12 + 4, 6 4, 2 + 9) = (16, 2, 11)

u ×v = ( 1 1, 3, 2 ) × ( 3, 2, 1 4 ) = (3 4 2 (2), (2) (3) 1 4, 1 2 (3) 3) = (12 + 4, 6 4, 2 + 9) = (16, 2, 11)

The length of this vector will now be the area of the parallelogram. You find the length like this:
162 + 22 + 112 = 256 + 4 + 121 = 381 19.5

162 + 22 + 112 = 256 + 4 + 121 = 381 19.5

The area of the parallelogram is approximately equal to 19.5.

Example 2

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.

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