Vectors
Visualization and mathematical definitions.
Discover Bridges
Study Vectors
What is a vector?
In physics we distinguish scalar and vector quantities. Scalar quantities are characterized by a numerical value (with unit). Vector quantities are given by a numerical value (with unit) and a direction. Vector quantities are often represented by an arrow (vector) the length of which represents the numerical value. The Analytic Geometry uses vectors for the description of e.g.
geometric objects (lines, planes, triangles etc.)
geometric operations (rotations, reflections etc.)
Vectors in the plane
Plane: two coordinate axes (e.g. \(x\)- and \(y\)-axis) intersecting in the zero-point (origin)
A vector in this plane is an arrow of which only the elongation in the direction of the \(x\)- and \(y\)-axes is known.
Example 1. \(\mathbf{v}:=\begin{pmatrix} 3\\-2\end{pmatrix}\) is a vector that starts at an arbitrary point and goes \(3\) steps in the direction of the \(x\)-axis and \(-2\) steps in the direction of the \(y\)-axis.
Vectors in \(3\)-dimensionsal space
\(3\)-dim space: three coordinate axes (e.g. \(x_1\)-, \(x_2\)- and \(x_3\)-axis) intersecting in the zero-point (origin)
A vector in this space is an arrow of which only the elongation in the direction of the \(x_1\)-, \(x_2\)- and \(x_3\)-axes is known.
Example 2. \(\mathbf{v}:=\begin{pmatrix} 1\\3\\2\end{pmatrix}\) goes 1 step in the direction of the \(x_1\)-axis, 3 steps in the direction of the \(x_2\)-axis and 2 steps in the direction of the \(x_3\)-axis.
Vectors in \(\mathbb{R}^n\)
Definition 3. Let \(n\in \mathbb{N}\). Every object \(\mathbf{v}\) of the form \[\mathbf{v}= \begin{pmatrix} v_1\\v_2\\ \vdots \\ v_n \end{pmatrix} ~~ \text{ mit }v_1,v_2,\ldots,v_n\in\mathbb{R}\] is called a real vector. The set of all such vectors is denoted by \(\mathbb{R}^n\). The entries \(v_1,v_2,\ldots,v_n\) are called components.
In different lectures you may see different notation for vectors: \(\mathbf{v}\), \(\vec{v}\), \(\underline{v}\) etc.
Exercise 4. Which of the given vectors are equal? Give their representations with coordinates.
Points and position vectors
Usually, a point \(P\) is given as a tuple \((v_1,v_2,\ldots,v_n)\). Such a point can also be represented by the corresponding vector \[\mathbf{p} = \begin{pmatrix} v_1\\v_2\\ \vdots \\ v_n \end{pmatrix}\, .\] In this case, we call \(\mathbf{p}\) the position vector of \(P\).
Definition 5. Let \(O\) be the origin (of the coordinate system). If \(P\) is a point, then the vector going from \(O\) to \(P\) is called the position vector of \(P\). We often identify points with their position vectors.
Vectors between two points
Let \(A\) and \(B\) be two points. The vector \(\mathbf{v}_{AB}\) going from \(A\) to \(B\) can be obtained by subtracting the position vector of \(A\) from the position vector of \(B\).
Example 6. The vector going from \(A:=(2,4,-6)\) to \(B:=(3,-1,9)\) is \[\mathbf{v}_{AB} = \begin{pmatrix} 3 \\ -1 \\ 9 \end{pmatrix} - \begin{pmatrix} 2 \\ 4 \\ -6 \end{pmatrix} =\begin{pmatrix} 1 \\ -5 \\ 15 \end{pmatrix}\]
Length of a vector
Definition 7. Consider any vector \(\mathbf{v} = \begin{pmatrix} v_1\\v_2\\ \vdots \\ v_n \end{pmatrix}\in \mathbb{R}^n\). Then the length of this vector is \[\| \mathbf{v} \| := \sqrt{v_1^2 + v_2^2 + \ldots + v_n^2}\, .\]
Example 8. The length of the vector \(\mathbf{v}:=\begin{pmatrix} 4\\-3 \end{pmatrix}\) is \(\|\mathbf{v}\| = \sqrt{4^2 + (-3)^2} = \sqrt{25} = 5\).
Discuss your questions by typing below.
Solve the WeBWorK Exercise
The data for the interactive network on this webpage was generated with pntfx Copyright Fabian Gabel and Julian Großmann. pntfx is licensed under the MIT license. Visualization of the network uses the open-source graph theory library Cytoscape.js licensed under the MIT license.