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

Question

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.

Answer 1

1. Why a Column of Ones is Added

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

A rectangle is a parallelepiped generated by the three vectors used.