The area of a triangle formed by vectors cross products

Sue writes,

Hi, I visited your website and think it is great! However I am having trouble with a question related to The Vector area of a triangle. The question is:

Three Vectors u, v, and w have a common initial point. Their endpoints form a triangle. Prove that the magnitude of the vector:

1/2(u×v + v×w + w×u)

Is equal to the area of the triangle. Such a vector is called the vector area of the triangle.

I would appreciate any help you could give me. Thank you very much!

First, remember that |a×b| = |a| |b| (sin θ)

(For an explanation, see Cross Product)

Let A be the endpoint of vector u, B be the endpoint of vector v, and C be the endpoint of vector W.

Then the vector from A to B is v-u, and the vector from A to C is w-u.

So (1/2) | (v-u) × (w-u) | is the area of the triangle.  (That's because the magnitude of the cross-product is equal to the area of the parallelogram determined by the two vectors, and the area of the triangle is one-half the area of the parallelogram.)

(v-u) × (w-u) = v×w - v×u - u×w + u×u

The cross product of a vector with itself is zero, and a×b = -b×a, so

(v-u) × (w-u) = v×w + u×v + w×u

which means that

(1/2) | (v-u) × (w-u) | = (1/2) | u×v + v×w + w×u | = area of the triangle.

Related Pages in this website

Triangle Area Using Vectors, part 1

Triangle Area using Determinant

Heron's formula for the area of a triangle, if all you know is the lengths of its sides.

Vectors -- explanation of Dot Product, Cross Product, and several vector identities using these products

Triple Product -- a·(b×c) is a scalar representing the "signed volume" of a parallelepiped

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