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I know that the area of a triangle is $\frac{1}{2}bh$ but today I learnt that to computing the area of a triangle using coordinates: $$ \text { Area of } \triangle A B C=\left|\frac{1}{2} \left| \begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right|\right|=\frac{1}{2}\left|\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]\right| $$

I came to know that this $\left|\frac{1}{2} \left| \begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right|\right|$ is called matrix. How did we get this matrix $\left|\frac{1}{2} \left| \begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right|\right|$? I am not able to understand this.

J. W. Tanner
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  • Do you know anything about determinants of matrices and their properties? – Misha Lavrov Oct 24 '21 at 14:24
  • If not, then the best explanation is probably "that matrix determinant is shorthand for the product of $x_i$'s and $y_j$'s on the right" and you can try to get that product from various differences of areas of right triangles and rectangles. – Misha Lavrov Oct 24 '21 at 14:25
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    Are you familiar with the cross product? That is, we have $$|\vec a\times\vec b|=|\vec a||\vec b|\sin\theta,$$ where $\theta$ is the angle between vectors $\vec a$ and $\vec b$. If $\vec a$ and $\vec b$ are two sides of a triangle, and $|\vec a|$ is the base of the triangle, then $|\vec b|\sin\theta$ is the height of the triangle. – Andrew Chin Oct 24 '21 at 14:27
  • Cf. this question – J. W. Tanner Oct 24 '21 at 14:31

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