# Векторы

## Содержание

* [#oboznacheniya](#oboznacheniya "mention")
* [#koordinaty-vektora](#koordinaty-vektora "mention")
* [#lineinaya-kombinaciya-vektorov](#lineinaya-kombinaciya-vektorov "mention")
* [#parallelnost-vektorov](#parallelnost-vektorov "mention")
* [#skalyarnoe-proizvedenie](#skalyarnoe-proizvedenie "mention")
* [#dlina-vektora](#dlina-vektora "mention")
* [#ugol-mezhdu-vektorami](#ugol-mezhdu-vektorami "mention")

## Обозначения

$$\vec{a}{a\_x,a\_y,a\_z}$$ — вектор $$a$$ с координатами $$a\_i$$;

$$\overrightarrow{AB}$$ — вектор с началом в точке $$A$$ и концом в точке $$B$$.

***

## Координаты вектора

$$
\large
\overrightarrow{AB}
{B\_x-A\_x;\~B\_y-A\_y;\~B\_z-A\_z}
$$

***

## Линейная комбинация векторов

$$
\large
m\cdot\vec{a}+k\cdot\vec{b}=\vec{c},

```\text{где}~~
c_i=m\cdot a_i + k \cdot b_i
$$

***

## Параллельность векторов

### Колинеарные векторы

$$
\large 
\vec{a} \parallel k\cdot \vec{a}
,~~k\in\R
$$

### Сонаправленные векторы

$$
\large \vec{a}
\upuparrows k\cdot \vec{a},
~~k>0
$$

### Противоположно направленные векторы

$$
\large \vec{a} \uparrow\downarrow k\cdot \vec{a},~~k<0
$$

***

## Скалярное произведение

$$
\large \vec{a}\cdot\vec{b}=
a_x\cdot b_x+
a_y\cdot b_y+
a_z\cdot b_z
$$

***

## Длина вектора

$$
\large 
\left|\vec{a}\right|=
\sqrt{\vec{a}^2}
$$

***

## Угол между векторами

$$
\large
\cos{\angle\left(\vec{a},\vec{b}\right)}=
\dfrac{\vec{a}\cdot\vec{b}}
{\left|\vec{a}\right|\cdot\left|\vec{b}\right|}
$$
```


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