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

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

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). □

Comments

Post a Comment

Popular posts from this blog

Open sets and closed sets in \(\mathbb{R}^n\)

-
Image
Open sets In \(\mathbb{R}\), we have the notion of an open interval such as \((a, b) = \{x \in \mathbb{R} | a < x < b\}\). We want to extend this idea to apply to \(\mathbb{R}^n\). We also introduce the notions of bounded sets and closed sets in \(\mathbb{R}^n\). Recall that the \(\varepsilon\)-neighbor of a point \(x\in\mathbb{R}^n\) is defined as \(N_{\varepsilon}(x) = \{y \in \mathbb{R}^n | d(x, y) < \varepsilon \}\) where \(d(x,y)\) is the distance between \(x\) and \(y\). Definition (Open set) A subset \(U\) of \(\mathbb{R}^n\) is said to be an  open set  if the following holds: \[\forall x \in U ~ \exists \delta > 0 ~ (N_{\delta}(x) \subset U).\tag{Eq:OpenSet}\] That is, for every point in an open set \(U\), we can always find an open ball centered at that point, that is included in \(U\). See the following figure. Perhaps, it is instructive to see what is  not  an open set. Negating (Eq:OpenSet), we have \[\exists x \in U ~ \forall \delta > 0 ~ (N_{\delta}(x) \not
Read more

Euclidean spaces

-
Image
We would like to study multivariate functions (i.e., functions of many variables), continuous multivariate functions in particular. To define continuity, we need a measure of "closeness" between points. One measure of closeness is the Euclidean distance. The set \(\mathbb{R}^n\) (with \(n \in \mathbb{N}\)) with the Euclidean distance function is called a Euclidean space. This is the space where our functions of interest live. The real line is a geometric representation of \(\mathbb{R}\), the set of all real numbers. That is, each \(a \in \mathbb{R}\) is represented as the point \(a\) on the real line. The coordinate plane , or the \(x\)-\(y\) plane , is a geometric representation of \(\mathbb{R}^2\), the set of all pairs of real numbers. Each pair of real numbers \((a, b)\) is visualized as the point \((a, b)\) in the plane. Remark . Recall that \(\mathbb{R}^2 = \mathbb{R}\times\mathbb{R} = \{(x, y) | x, y \in \mathbb{R}\}\) is the Cartesian product of \(\mathbb{R}\) with i
Read more

Newton's method

-
Image
Newton's method (the Newton-Raphson method) is a very powerful numerical method for solving nonlinear equations.  Suppose we'd like to solve a non-linear equation \(f(x) = 0\), where \(f(x)\) is a (twice) differentiable non-linear function. Newton's method generates a sequence of numbers \(c_1, c_2, c_3, \cdots\) that converges to a solution of the equation. That is, if \(\alpha\) is a solution (i.e., \(f(\alpha) = 0\)) then,  \[\lim_{n\to\infty}c_n = \alpha,\] and this sequence \(\{c_n\}\) is generated by a series of linear approximations of the function \(f(x)\). Theorem (Newton's method) Let \(f(x)\) be a function that is twice differentiable on an open interval \(I\) that contains the closed interval \([a, b]\) (i.e., \([a,b]\subset I\)) and satisfy the following conditions: \(f(a) < 0\) and \(f(b) > 0\); For all \(x\in [a, b]\), \(f'(x) > 0\) and \(f''(x) > 0\). Let us define the sequence \(\{c_n\}\) by \[ \begin{eqnarray} c_1 &
Read more