Why does the half of the absolute value of a matrix formed with its coordinates give

[BlueRope]

Why does the half of the absolute value of a matrix formed with its coordinates give the area of a triangle?


I don't see any similarity between that and the heron's formula.

 

[dx]

You mean determinant.

 

[BlueRope]

Yeah, why is that?

 

[Jim Kata]

The reason is because half the area of a parallelogram is the area of a triangle. Say you have k linearly independent vectors, the volume of the parallelepiped they span is given by the formula:V(x_1,...,x_k)= \sqrt{det(X^{tr}X)}. This formula is easiest to understand in 3 dimensions. Say you have two vectors a and b which are elements of \mathbb{R}^3 then the volume formula says that V(a,b)^2=det\begin{bmatrix}<br /> \|a\|^2 &amp; \langle a,b \rangle \\<br /> \langle b,a \rangle &amp; \|b\|^2 \\<br /> \end{bmatrix} = \|a\|^2\|b\|^2-\langle a,b \rangle^2 = \|a\|^2\|b\|^2(1-\cos^2(\theta)) = \|a\|^2\|b\|^2\sin^2(\theta) which is just the area of the parallelogram spanned by a and b. This is the well known formula that magnitude of the cross product of the two vectors is the area of the parallelogram spanned by the two vectors.