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 $$V(a,b)^2=det\begin{bmatrix} \|a\|^2 & \langle a,b \rangle \\ \langle b,a \rangle & \|b\|^2 \\ \end{bmatrix} = \|a\|^2\|b\|^2-\langle a,b \rangle^2 = \|a\|^2\|b\|^2(1-\cos^2(\theta)) = \|a\|^2\|b\|^2\sin^2(\theta)$$ 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.

 

[blue_raver22]

Can you please provide more context or explanation for your statement? Also, what do you mean by "half of the absolute value of a matrix formed with its coordinates"? A matrix is a mathematical object used to represent linear transformations and has no inherent connection to the concept of area. It would be helpful to have more information in order to provide a thorough response.