I asked this in MSE, it flashed and disappeared.

Let $V_n$ be the volume on the set of polytopes in $\mathbb R^n$, defined axiomatically, i.e. a functional, that assigns to each polytope $P\subseteq\mathbb R^n$ a real number $V_n(P)\ge 0$ in such a way that the following conditions are fulfilled:

1. For the unit hypercube $C\subseteq\mathbb R^n$ $$ V_n(C)=1, $$

2. If $P\cap Q=\varnothing$, then $$ V_n(P\cup Q)=V_n(P)+V_n(Q), $$

3. If $P$ is made from $Q$ by a motion, then $$ V_n(P)=V_n(Q). $$ Question:

Is there a simple way to prove that the $n$-th volume of a parallelepiped $P_{a_1,...,a_n}$ spanned by $n$ vectors $a_1,...,a_n$ is the poduct of the $n-1$-th volume of $P_{a_1,...,a_{n-1}}$ and the height of $P_{a_1,...,a_n}$ with respect to the base $P_{a_1,...,a_{n-1}}$, $$ V_n(P_{a_1,...,a_n})=V_{n-1}(P_{a_1,...,a_{n-1}})\cdot \left|\text{pr}_{\{a_1,...,a_{n-1}\}^\perp}(a_n)\right| $$ ?

(Here $\text{pr}_{\{a_1,...,a_{n-1}\}^\perp}(a_n)$ is the orthogonal projection of $a_n$ to the orthogonal complement $\{a_1,...,a_{n-1}\}^\perp$ of $\{a_1,...,a_{n-1}\}$, i.e. the height of $P_{a_1,...,a_n}$.)

P.S. This is not for math reasearch, this is for teaching.

I asked this in MSE, it flashed and disappeared.

Let $V_n$ be the volume on the set of polytopes in $\mathbb R^n$, defined axiomatically, i.e. a functional, that assigns to each polytope $P\subseteq\mathbb R^n$ a real number $V_n(P)\ge 0$ in such a way that the following conditions are fulfilled:

1. For the unit hypercube $C\subseteq\mathbb R^n$ $$ V_n(C)=1, $$

2. If $P\cap Q=\varnothing$, then $$ V_n(P\cup Q)=V_n(P)+V_n(Q), $$

3. If $P$ is made from $Q$ by a motion, then $$ V_n(P)=V_n(Q). $$ Question:

Is there a simple way to prove that the $n$-th volume of a parallelepiped $P_{a_1,...,a_n}$ spanned by $n$ vectors $a_1,...,a_n$ is the poduct of the $n-1$-th volume of $P_{a_1,...,a_{n-1}}$ and the height of $P_{a_1,...,a_n}$ with respect to the base $P_{a_1,...,a_{n-1}}$, $$ V_n(P_{a_1,...,a_n})=V_{n-1}(P_{a_1,...,a_{n-1}})\cdot \left|\text{pr}_{\{a_1,...,a_{n-1}\}^\perp}(a_n)\right| $$ ?

(Here $\text{pr}_{\{a_1,...,a_{n-1}\}^\perp}(a_n)$ is the orthogonal projection of $a_n$ to the orthogonal complement $\{a_1,...,a_{n-1}\}^\perp$ of $\{a_1,...,a_{n-1}\}$, i.e. the height of $P_{a_1,...,a_n}$.)

P.S. This is not for math reasearch, this is for teaching.

I asked this in MSE, it flashed and disappeared.

Let $V_n$ be the volume on the set of polytopes in $\mathbb R^n$, defined axiomatically, i.e. a functional, that assigns to each polytope $P\subseteq\mathbb R^n$ a real number $V_n(P)\ge 0$ in such a way that the following conditions are fulfilled:

1. For the unit hypercube $C\subseteq\mathbb R^n$ $$ V_n(C)=1, $$

2. If $P\cap Q=\varnothing$, then $$ V_n(P\cup Q)=V_n(P)+V_n(Q), $$

3. If $P$ is made from $Q$ by a motion, then $$ V_n(P)=V_n(Q). $$ Question:

Is there a simple way to prove that the $n$-th volume of a parallelepiped $P_{a_1,...,a_n}$ spanned by $n$ vectors $a_1,...,a_n$ is the poduct of the $n-1$-th volume of $P_{a_1,...,a_{n-1}}$ and the height of $P_{a_1,...,a_n}$ with respect to the base $P_{a_1,...,a_{n-1}}$, $$ V_n(P_{a_1,...,a_n})=V_{n-1}(P_{a_1,...,a_{n-1}})\cdot \left|\text{pr}_{\{a_1,...,a_{n-1}\}^\perp}(a_n)\right| $$ ?

(Here $\text{pr}_{\{a_1,...,a_{n-1}\}^\perp}(a_n)$ is the orthogonal projection of $a_n$ to the orthogonal complement $\{a_1,...,a_{n-1}\}^\perp$ of $\{a_1,...,a_{n-1}\}$, i.e. the height of $P_{a_1,...,a_n}$.)

I asked this in MSE, it flashed and disappeared.

Let $V_n$ be the volume on the set of polytopes in $\mathbb R^n$, defined axiomatically, i.e. a functional, that assigns to each polytope $P\subseteq\mathbb R^n$ a real number $V_n(P)\ge 0$ in such a way that the following conditions are fulfilled:

1. For the unit hypercube $C\subseteq\mathbb R^n$ $$ V_n(C)=1, $$

2. If $P\cap Q=\varnothing$, then $$ V_n(P\cup Q)=V_n(P)+V_n(Q), $$

3. If $P$ is made from $Q$ by a motion, then $$ V_n(P)=V_n(Q). $$ Question:

Is there a simple way to prove that the $n$-th volume of a parallelepiped $P_{a_1,...,a_n}$ spanned by $n$ vectors $a_1,...,a_n$ is the poduct of the $n-1$-th volume of $P_{a_1,...,a_{n-1}}$ and the height of $P_{a_1,...,a_n}$ with respect to the base $P_{a_1,...,a_{n-1}}$, $$ V_n(P_{a_1,...,a_n})=V_{n-1}(P_{a_1,...,a_{n-1}})\cdot \left|\text{pr}_{\{a_1,...,a_{n-1}\}^\perp}(a_n)\right| $$ ?

(Here $\text{pr}_{\{a_1,...,a_{n-1}\}^\perp}(a_n)$ is the orthogonal projection of $a_n$ to the orthogonal complement $\{a_1,...,a_{n-1}\}^\perp$ of $\{a_1,...,a_{n-1}\}$, i.e. the height of $P_{a_1,...,a_n}$.)

P.S. This is not for math reasearch, this is for teaching.