Position vectors

We can use vectors to represent positions in space. By definition, (geometric) vectors don't have positions in space (or their positions are irrelevant). Nevertheless, we can always choose a representative vector from its equivalence class such that the source is nailed down to the origin. Such a vector is called a position vector, as its target corresponds to a point in space.

Example. Let us find the equation of the straight line in ${\mathbb{R}}^2$ that goes through the two points $(1,2)$ and $(3,4)$. The position vectors are $\mathbf{a}=(1,2)$ and $\mathbf{b}=(3,4)$. $\mathbf{b}-\mathbf{a}$ is a vector which is the direction of the line from =(\mathbf{a}\) to $\mathbf{b}$. Let $\mathbf{r}$ be the position vector of any point on the desired line. Then $\mathbf{r}-\mathbf{a}$ is a vector which is the direction of the line from $\mathbf{a}$ to $\mathbf{r}$, which also goes through the point $\mathbf{b}$. Thus we should have

$$\mathbf{r}-\mathbf{a}=\lambda (\mathbf{b}-\mathbf{a})$$

for some $\lambda \in \mathbb{R}$. Thus we have

$$\mathbf{r}=\mathbf{a}+\lambda (\mathbf{b}-\mathbf{a})$$

as the general equation for the points on the line that goes through $\mathbf{a}$ and $\mathbf{b}$.

If $\lambda =0$, then $\mathbf{r}=\mathbf{a}$. If $\lambda =1$, then $\mathbf{r}=\mathbf{b}$. If $0<\lambda <1$, then $\mathbf{r}$ is somewhere between $\mathbf{a}$ and $\mathbf{b}$. □

Example. Let $\mathbf{c}\in {\mathbb{R}}^n$ be a constant vector. The set of position vectors defined by

$$P=\{\mathbf{r}\in {\mathbb{R}}^n|⟨\mathbf{r}-\mathbf{c},\mathbf{c}⟩=0\}$$

represents the ``plane'' that contains the point $\mathbf{c}$ and is perpendicular to the vector $\mathbf{c}$. In ${\mathbb{R}}^2$, this ``plane'' is actually a line. In ${\mathbb{R}}^3$, this is indeed a plane. In ${\mathbb{R}}^n$ with $n>3$, this ``plane'' is usually called a hyperplane. Find out why this is so named. □

Example. Let $\mathbf{c}\in {\mathbb{R}}^n$ be a fixed position vector and $r\in \mathbb{R}$ a positive constant. The set of position vectors defined by

$$S=\{\mathbf{x}\in {\mathbb{R}}^n\mid \Vert \mathbf{x}-\mathbf{c}\Vert =r\}$$

defines an $n$-sphere centered at $\mathbf{c}$ with radius $r$. If $n=2$, this ``sphere'' is a circle. If $n=3$, this is indeed a sphere. If $n>3$, we just call it an $n$-sphere or a hypersphere. 

Similarly,

$$B=\{\mathbf{x}\in {\mathbb{R}}^n\mid \Vert \mathbf{x}-\mathbf{c}\Vert \le r\}$$

defines an $n$-ball centered at $\mathbf{c}$ with radius $r$. If $n=2$, this is a disk. If $n=3$, this is indeed a ball. Balls play an important role in topology (a study of shapes based on the ``closeness'' of points). □