Position vectors

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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

\[P = \{\mathbf{r} \in \mathbb{R}^n | \braket{\mathbf{r}-\mathbf{c}, \mathbf{c}}  = 0\}\]

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
\[S = \{\mathbf{x} \in \mathbb{R}^n \mid \|\mathbf{x} - \mathbf{c}\| = r\}\]
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,
\[B = \{\mathbf{x} \in \mathbb{R}^n \mid \|\mathbf{x} - \mathbf{c}\| \leq r\}\]
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). □

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