Questions about the dot products of a unit cube

Question

1. I want to find the dot products(It should have 16 dot products.) of a unit cube's diagonal lines.

P.S. unit cube is a cube where all the lengths of 12 edges of the cube are 1.

2. It should have 16 dot products, right?

3. Moreover, what is the projection of a vector on a plane in $R^n$?

Answer 1

For your second point: First, pick an inner product $\langle\cdot,\cdot\rangle$ on $\mathbb{R}^n$ and choose an orthonormal basis $(e_1,e_2)$ of your plane $P\subset\mathbb{R}^n$, i.e. $e_1,e_2\in\mathbb{R}^n$ are two vectors with $\|e_1\|=\|e_2\|=1$ and $\langle e_1,e_2\rangle=0$.

Then, given any vector $v\in\mathbb{R}^n$, its projection $\hat v$ to $P$ is given by $$\hat v = \langle v,e_1\rangle e_1 + \langle v, e_2\rangle e_2.$$

In the special case of $n=3$, there also exists a special vector $e_3\in\mathbb{R}^3$ called the unit normal vector to $P$, which satisfies $\|e_3\|=1$ and $\langle e_1,e_3\rangle = \langle e_2,e_3\rangle = 0$. Using this vector, the projection of $v$ to $P$ can also be calculated as $$\hat v = v - \langle v,e_3\rangle e_3.$$