Vector Operations

addition and subtraction

You can add and subtract vectors (in the same dimension).

${\vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} }$ and ${\vec{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} }$ .

Then

${\vec{a} + \vec{b} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix}}$

Simililarly,

${\vec{a} - \vec{b} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} - \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 - b_1 \\ a_2 - b_2 \end{pmatrix}}$

simulation - PHET

scalar multiplication

${k \cdot \vec{a} = k \cdot \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} = \begin{pmatrix} ka_1 \\ ka_2 \end{pmatrix} }$

Geometrically, you are expanding or compressing the vector.

If the scalar is negative, the direction the vector is pointing in is reversed.

TODO - images

base vectors

A vector $\textbf{i}$, where ${\textbf{i} = \vec{i} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} }$ is a unit vector in direction of the x-axis.

Similary,

${\textbf{j} = \vec{j} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} }$

${\textbf{k} = \vec{k} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} }$

Using addition and scalar multiplication, it is now possible to express

${\vec{a} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} }$ as ${\textbf{a} = 3\textbf{i} + \textbf{j} + 2\textbf{k}}$.

You, will find this notation in many older books where typesetting column vectors was not possible. (see A-Level and IB)

other multiplication of vectors


(c) 2019 sebastian.williams[at]sebinberlin.de - impressum und datenschutz - Powered by MathJax & XMin & HUGO & jsxgraph & mypaint