Volume of general parallelepiped.

Question

$\newcommand{\b}{\mathbf}$Following two statements were given in the book to motivate the reader about the formal definition of determinants but I only got demotivated and frustrated by these statements.

The parallelepiped determined by the vectors $\b v_1 ,\b v_2 , . . . , \b v_n$ can be defined as the collection of all vectors $\b v ∈ \Bbb R^n$ that can be represented as $\b v = t_1\b v_1 + t_2\b v_2 + . . . + t_n\b v_n$ , $0 ≤ t_k ≤ 1\ ∀k = 1, 2, . . . , n.$

And

We know, that for dimensions $2$ and $3$ “volume” of a parallelepiped is determined by the base times height rule: if we pick one vector, then height is the distance from this vector to the subspace spanned by the remaining vectors, and the base is the $(n − 1)$-dimensional volume of the parallelepiped determined by the remaining vectors.

In first statement I didn't understand why author says that a parrellepiped is defined by infinitely many vectors. Don't we only need position vectors of the vertices of parrallelepiped to define it ? Like in $\Bbb R^2$, $\{ (1,0)^T, (0,1)^T, (1,1)^T, (0,0)^T\}$ define a parrallelepiped (parallelogram).

In the second statement, it would be hyperbole if I say I understand anything. More specifically I did not understand "if we pick one vector, then height is the distance from this vector to the subspace spanned by the remaining vectors", what does this even mean ? Distance between a vector and a vector space ? I think my lack of background in geometry is bitting me.

Answer 1

You need three vectors to determine a parallelepiped. This is because you know that one of the vertices is at $0$, so if you know the positions of the remaining $3$ "nearest" vertices, the rest is determined by the parallel-ness of a parallelepiped. I will refer to these as the "defining vectors".

So let's say we have the three vectors $\textbf{a}, \textbf{b}, \textbf{c}$. Then is the parallelepiped precisely the set $\{\textbf{a}, \textbf{b}, \textbf{c}\}$? Clearly this isn't true, because $\{\textbf{a}, \textbf{b}, \textbf{c}\}$ is just a collection of three vectors, it's not a parallelepiped. What you need is the entire volume - every point it contains. For example, you need the point at the origin: $0 = 0\textbf{a}+ 0\textbf{b}+ 0\textbf{c}$. How about the vertex on the "base" furthest to the right? You can see that that point is given by $1\textbf{b}+ 1\textbf{c}$. What about the centre of the parallelepiped? That point is $0.5\textbf{a}+0.5 \textbf{b}+0.5 \textbf{c}$. In this way, we see that the $\textit{collection of all points in the parallelepiped}$ is given by $\{t_1\textbf{a}+t_2 \textbf{b}+t_3 \textbf{c}: 0 ≤ t_i ≤ 1\}$.

Now for your second question: Paragraph 2.

The volume is given by $base\times height$, in other words, the area of the shaded region multiplied by $h$. We want to figure out what $h$ is.

First we "pick vector $\textbf{a}$". Then you have two defining vectors, $\textbf{b}$ and $\textbf{c}$, left over. The subspace spanned by $\textbf{b}$ and $\textbf{c}$ is $\{t_2\textbf{b}+t_3\textbf{c}:t_i\in \mathbb{R}\}$. But from the above definition of a parallelepiped, we see that the "base" of the parallelepiped is in fact $\{0\textbf{a} + t_2\textbf{b}+t_3\textbf{c}:0≤t_i≤1\} = \{t_2\textbf{b}+t_3\textbf{c}:0≤t_i≤1\} \subset \{t_2\textbf{b}+t_3\textbf{c}:t_i\in \mathbb{R}\}$. This means the height of the parallelepiped, i.e. the "distance from the top to the base" is equal to the "distance from the top to the subspace spanned by the other two defining vectors".

As for the "top", we see that the height of the parallelepiped is determined by the position of $\textbf{a}$. Hence the "distance from the top to the subspace spanned by the other two defining vectors" is in fact the "distance from $\textbf{a}$ to the subspace spanned by $\textbf{b}$ and $\textbf{c}$".

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Note: The easiest way to think about "distance from a vector to a subspace" is to think of it as the length of the line perpendicular to subspace which passes through the endpoint of the vector. Here's an example. The distance from $\textbf{b}$ to the yellow subspace is the length of $\textbf{b}-\textbf{p}$, becuase $\textbf{b}-\textbf{p}$ is perpendicular to the yellow subspace, and it passes through the end point of $\textbf{b}$.

Citations: First image is from wikipedia, second image is from ms.uky.edu