Vector algebra

Overview: Vector algebra

This article gives a brief explanation of vectors and vector algebra.
A scalar value is just a numeric value. In contrast thereto, a vectorial value has a direction. Examples for vectors are all forces and the velocity, which is directed to the driving direction. Vectors are usually denoted with an arrow, for example $\vec{\mathbf{a}}$ . A three-dimensional vector for example is

$\vec{\mathbf{a}}= \left[\begin{array}{cccc} a_x\\ a_y\\ a_z \end{array}\right]$

where $a_x$ , $a_y$ and $a_z$ describe the components of the vector in x-, y- and z-direction. For graphical representations arrows are used that show the direction of the vector and whose length equals its magnitude. The magnitude of a vector is defined as

$|\vec{\mathbf{a}}|=\sqrt{a_x^2+a_y^2+a_z^2}$

As the presented notation is comparatively sophisticated $|\vec{\mathbf{a}}|$ is often reduced to $a$ . The vector $-\vec{\mathbf{a}}$ has the same magnitude as $\vec{\mathbf{a}}$ but is directed opposite. Two or more vectors are called equal if they have the same direction and the same magnitude. The zero vector $\vec{\mathbf{0}}$ is a special vector with undefined direction and magnitude zero.

Depending on their properties vectors can be characterized as follows:

Free vector: Free vectors can be moved arbitrarily in space. They are not constrained to a fixed point. So similar vectors can be aligned using translation.
Constrained vector: Constrained vectors can not be moved because they are constrained to a fixed point in space. A force for example forces act in a certain point and in a certain direction.
Position vector: Position vectors usually start in the origin of a given coordinate system and point at a certain position of a point. In this sense they are constrained vectors.

The following subarticles describe the unit vector and the basic arithmetic operations including dot and cross product:

Example: Movement of a flying object

Consider a flying object at time $t$ . This object not only has a current velocity $v(t)$ but also a direction of motion. Thus the velocity of the object is a vector $\vec{\mathbf{v}}(t)$ . The corresponding distance then arises out of the product of the velocity vector and the time:

$\vec{\mathbf{s}}(t)=\vec{\mathbf{v}}(t)\cdot t$