%path = "maths/vectors" %kind = chindnum["texts"] %level = 11
A multidimensional vector can be seen as independently choosing (value) from more variables.
The values (number+unit) must be addable independently.
The units are the unit vectors. Together they form the basis and are therefore also called basis vectors.
The choice from one variable is a vector, too, a one-dimensional vector.
The whole vector can be multiplied by a number, the scalar, and yields a vector again.
Example:
If I go into a shop, then the products there are my vector space (coordinate system, CS) and my shopping basket is a vector, i.e. a fixing of the value (how much?) of each variable (here product).
If my wife went shopping, too, then the baskets add up independently at home, i.e. milk + milk, butter + butter, …
A matrix transforms a vector from one coordinate system to a vector of another coordinate system. Therefore we learn first about vectors. The matrix comes into play, when we want to change from one coordinate system to another.
Example
As column of numbers \(\vec{x}=\begin{pmatrix}x_1\\x_2\end{pmatrix}\). The unit vectors, i.e. what the rows mean, one specifies separately.
Written explicitly with units: \(\vec{x}=x_1\vec{e_1}+x_2\vec{e_2}\) (3 milk + 5 butter). If without arrow, then the superscript index normally mean the scalar (number) and the subscript index the unit (dimension, direction): \(x=x^1e_1+x^2e_2\).
Notation is not the vector itself.
Apart from addition there are two other important vector operations.
dot-product (scalar product). It yields a number (scalar) that represents the dependence or with how little independence one can choose values.
Orthogonal vectors result in 0 (no dependence).
For parallel vectors it is the product of the lengths. The length of a vector \(\vec{x}\) is thus \(\sqrt{\vec{x}\vec{x}}\) The length is denoted as \(|\vec{x}|\) or simply \(x\).
\(\vec{x_o}=\frac{\vec{x}}{x}\) is the unit vector (length 1 in the direction of \(\vec{x}\))
The dot-product defines an angle between two vectors: \(\cos\alpha = \frac{\vec{x}\vec{y}}{xy}\)
Vector product or cross product. For a dimension \(= 3\) it produces a vector orthogonal to \(\vec{x}\) and \(\vec{y}\) and of length equal to the area of the parallelogram created by the two vectors.
If \(\vec{x}\) and \(\vec{y}\) are two-dimensional, then only the \(\vec{e_3}\) component of \(\vec{x}\times\vec{y}\) is different from 0. It is \(\begin{vmatrix} x_1 & x_2 \\ y_1 & y_2 \end{vmatrix}= \begin{vmatrix} x_1 & y_1 \\ x_2 & y_2 \end{vmatrix}\). Compare this to: Determinant of 3 vectors in the 3D space are the volume of the parallelepiped created by the three vectors.