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)