I know that if we put vectors in rows and calculate the determinant we get the volume of a parallelogram.
I have read the following example:
Find the straight line equation that goes through the points $(1,2),(3,-5)\in \mathbb{R}^2$ using Determinant.
How should I approach it?
This means that the vector $\vec{v}=(1,2)-(3,-5)=(-2,7)$ is a parallel vector to the line. Let's call $A (1,2)$ and $B$ $(3,5)$. Now, for a random point $M(x,y)$ on the line, $\vec{AM}=(x-1,y-2)$ and $\vec{v}=(-2,7)$ are parallel (colinear), hence their determinant must be zero. So $7(x-1)+2(y-2)=0$. This way uses determinant, there are many other ways!