Vector area of a triangle
The vector area of Δ ABC is
(AB × AC) =
( BC × BA ) =
(CA × CB)
Proof:
Let n be the unit vector in the direction of AB × AC
Let Δ be the Area of the Δ ABC.
∴ Δ = (AB)(AC) sin A
| ∴ The vector area | = | Δ n | |
| = | (AB)(AC) (sin A)n
| ||
| = | |AB| |AC| (sin A)n |
||
| = | (AB × AC) |
Similarly, we can prove the vector area = (BC × BA)
= (CA × CB)