# Need help understanding computation of area and volume using Matrices and linear algebra?

I am currently studying linear algebra from Gilbert R Strang's book. I have a few questions regarding computing area using matrices.

In the book, for a triangle with coordinates: $$(x_1, y_1)$$, $$(x_2, y_2)$$, $$(x_3, y_3)$$

The area is given as 1/2*det($$\begin{matrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{matrix}$$)

1. Is the intuition behind adding the column of 1's just to make it into a square matrix?

Moving onto calculating volume of a rectangle with sides of length (r,s,t). The coordinate matrix will be: $$\begin{matrix} 0 & 0 & 0 \\ r & 0 & 0 \\ 0 & s & 0 \\ 0 & 0 & t \\ r & s & 0 \\ 0 & s & t \\ r & 0 & t \\ r & s & t \\ \end{matrix}$$

2. Going by the logic applied to compute area of triangle, will I need to add 5 columns of 1's in order to get the volume.

Because in the book, the author mentions that computing the determinant of the diagonal matrix A with diagonal entries r,s,t would give the volume as rst (which is true by the Lbh formula)

3. But, what I fail to understand is how does a computing the determinant of a subset of three vectors give me the volume?

Also if I use the three vectors (r,s,0) , (r, 0 ,t) and (0, s, t) to do the same, the value of the determinant is -2rst which is the incorrect volume. Do the three vectors chosen need to be perpendicular?

4. Basically I am having trouble wrapping my head around how you need all three coordinates to compute the area but only 3 coordinates (vectors?) to compute the volume despite there being 8 coordinates that bound the rectangle in 3D space.

The volume of the parallelepiped that is generated by the vectors $$u,v,w$$ is given by $$\det\begin{bmatrix} u_x&u_y&u_z\\ v_x&v_y&v_z\\ w_x&w_y&w_z \end{bmatrix}\tag1$$ The volume of the tetrahedron whose vertices are $$0,u,v,w$$ is $$\frac16$$ of the volume of the parallelepiped.
Furthermore, the volume of any pyramid is $$\frac13\times|\text{base}|\times\text{altitude}$$. By setting the $$z$$-coordinate of each vertex of the triangle to $$1$$, we have essentially set the altitude to $$1$$ (the dotted line in the rotating image) and the base to the triangle. Thus, the area of the triangle is $$3\times\text{volume}=\frac12\det\begin{bmatrix} u_x&u_y&1\\ v_x&v_y&1\\ w_x&w_y&1 \end{bmatrix}\tag2$$ 2. Rectangle With Given Measurements
A rectangle with edges $$r,s,t$$ is a parallelepiped generated by $$(r,0,0),(0,s,0),(0,0,t)$$, and so its volume is $$\det\begin{bmatrix}r&0&0\\0&s&0\\0&0&t\end{bmatrix}=rst\tag3$$ 3. Why the Determinant of Three Vectors Gives the Volume of a Rectangle