Suppose P and Q are two distinct points. Prove that there exists exactly one translation which sends P to Q

${\displaystyle \text{Let}over\to \left\{e_1\right\},over\to \left\{e_2\right\},over\to \left\{e_3\right\}\text{be standard unit vectors along the coordinate axes in}{R}^3.\text{Let S and T be the linear transformations defind in}{R}^3.\text{Show that if}}$
${\displaystyle S(over\to {\left\{e\right\}}_1)=T(over\to {\left\{e\right\}}_1),S(over\to {\left\{e\right\}}_2)=T(over\to {\left\{e\right\}}_2),S(over\to {\left\{e\right\}}_3)=T(over\to {\left\{e\right\}}_3)}$
then
${\displaystyle S(over\to \left\{x\right\})=T(over\to \left\{x\right\})}$
for any ${\displaystyle over\to \left\{x\right\}\in R^3}$

${\displaystyle \text{Let A be an}n\times n\text{matrix and suppose that}L:M_{\cap }\to M_{\cap }\text{is defined by}L\left(x\right)=AX,\text{for}X\in M_{\cap }.\text{Show that L is a linear transformation.}}$

To prove: ${\displaystyle D^{+}\subseteq Q^{+}}$
Given information:
${\displaystyle \text{Each}x\in D\text{is identified with}[x,e]\text{in Q}}$

Whether T is a linear transformation, that is ${\displaystyle T:C^1[-1,1]\to R^1}$ defined by ${\displaystyle T\left(f\right)=f^{\prime }\left(0\right)}$

Are partial derivatives of parametric surfaces always orthogonal?
For a surface $\mathbf{r}=\mathbf{r}(s,t)$ are the partial derivates ${\mathbf{r}}_s$ and ${\mathbf{r}}_t$ in general orthogonal?

Geometric Probability in Multiple Dimensions
I understand how Geometric Probability works in 1, 2, and 3 dimensions, but is it possible to do these problems in, say, 5 dimensions? For example,
Five friends are to show up at a party from 1:00 to 2:00 and are to stay for 6 minutes each. What is the probability that there exists a point in time such that all of the 5 friends meet?

I have not yet had the privilege of studying multivariable calculus, but I have made an educated guess about how to find the minimum or maximum of a function with two variables, for example, $x$ and $y$.

Since, in three dimensions, a minimum or maximum would be represented by a tangent plane with no slope in any direction, could I treat $y$ as a constant and differentiate $z$ with respect to $x$, then treat $x$ as a constant and differentiate with respect to $y$, and find the places where both of these two are equal to zero?

Sorry if this is just a stupid assumption... it may be one of those things that just seems correct but is actually wrong.