# Vectors

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## What are Vectors?

There are two main measures: scalar and vector. Scalar is a quantity that has size but hasn’t direction (for example, time, volume, temperature). Vector is a measure that has length and direction (for example, force, acceleration, velocity).

In Picture 1 the line AB is represented. It is known that point A is the beginning and point B is the end of the line. That’s why line AB is called the vector. Vector $$\overrightarrow{AB}$$ is directed from A to B.

This vector is denoted $$\overrightarrow{AB}$$ (read: «vector AB»). In pictures, vectors are represented by lines with arrows on the end. Vectors are denoted by Latin letters with arrows ($$\overrightarrow{a}$$, $$\overrightarrow{b}$$, $$\overrightarrow{c}$$) (Picture2).

A vector with length 0 is called a zero vector. A zero vector is denoted as $$\overrightarrow{0}$$. If point A is the beginning and the end of a zero vector, this vector can be denoted as $$\overrightarrow{AA}$$ too. A zero vector is represented as one point in pictures. A vector with length 1 is called a unit vector.

The module of the vector $$\overrightarrow{AB}$$ is a length of the line AB. The module of the vector $$\overrightarrow{AB}$$ is denoted $$|\overrightarrow{AB}|$$, and the module of the vector $$\overrightarrow{a}$$ is denoted $$|\overrightarrow{a}|$$. The module of the zero vector is equal 0: $$|\overrightarrow{0}|$$=0.

Vectors which lie on parallel lines or on the same line are called collinear vectors. A zero vector is collinear to all vectors. There are represented collinear vectors $$\overrightarrow{a}$$, $$\overrightarrow{b}$$ and $$\overrightarrow{c}$$ in Picture 3. Collinear vectors are denoted $$\overrightarrow{a}|$$$$|\overrightarrow{b}$$.

If collinear vectors are directed on the same side, they are called equally directed vectors and they are denoted as $$\overrightarrow{a}↑$$$$↑\overrightarrow{c}$$.  If collinear vectors have different directions, they are called reverse directed vectors and are denoted $$\overrightarrow{a}↑↓$$$$\overrightarrow{b}$$ (Picture 4). A zero vector isn’t equally or reverse directed to other vectors.

Reverse vectors with the same length are called inverse vectors. For example, vectors $$\overrightarrow{AB}$$ and $$\overrightarrow{BA}$$ are inverse vectors, $$\overrightarrow{AB}$$=$$-\overrightarrow{BA}$$  and  $$\overrightarrow{BA}$$=$$-\overrightarrow{AB}$$.
Vectors which have the same magnitude (but not zero) and have equal direction are called same vectors (Picture 5).
Consider vector $$\overrightarrow{a}$$ on the coordinate plane (Picture 6). Construct same vector to $$\overrightarrow{a}$$, which begins in the beginning of coordinate plane and ends in point A. The coordinates of the vector $$\overrightarrow{a}$$ are coordinates of point A. The coordinates of the vector $$\overrightarrow{a}$$ are denoted $$\overrightarrow{a}$$$$(x_{a};y_{a})$$. Same vectors have equal corresponding coordinates.

If two points, A$$(x_{1};y_{1})$$ and B$$(x_{2};y_{2})$$, which are the beginning and end of the vector $$\overrightarrow{AB}$$, respectively, correspond, then the numbers $$x_{2}-x_{1}$$ and $$y_{2}-y_{1}$$ are, respectively, first and second coordinates of vector $$\overrightarrow{AB}$$.

When vector $$\overrightarrow{a}$$ has coordinates $$x_{a}$$ and $$y_{a}$$, then $$|\overrightarrow{AB}|=$$$$\sqrt{x^{2}_{a}+y^{2}_{a}}$$.

Sum of two vectors is a vector.

Rule of triangle: construct vector $$\overrightarrow{AB}$$, equal to the vector $$\overrightarrow{a}$$, from arbitrary point A and vector $$\overrightarrow{BC}$$ from point B, equal to the vector $$\overrightarrow{b}$$. Vector $$\overrightarrow{AC}$$ is called sum of vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ (Picture 7). It is written as $$\overrightarrow{a}$$ + $$\overrightarrow{b}$$=$$\overrightarrow{AC}$$.
If coordinates of vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ correspond equally to $$(x_{a};y_{a})$$ and
$$(x_{b};y_{b})$$, then coordinates of the vector $$\overrightarrow{a}+$$$$\overrightarrow{b}$$ are equal to $$(x_{a}+x_{b};y_{a}+y_{b})$$.
Difference between two vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ is the vector $$\overrightarrow{c}$$, if the sum of this vector and vector $$\overrightarrow{b}$$ are equal to $$\overrightarrow{a}$$.
Construct vectors $$\overrightarrow{OA}$$ and $$\overrightarrow{OB}$$ correspond equally to vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ from arbitrary point O. Then, vector $$\overrightarrow{BA}$$ differs between $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ (Picture 8).

If coordinates of vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ are correspondingly equal to $$(x_{a};y_{a})$$ and
$$(x_{b};y_{b})$$, then coordinates of the vector $$\overrightarrow{a}-$$$$\overrightarrow{b}$$ are equal to $$(x_{a}-x_{b};y_{a}-y_{b})$$.

For any two vectors, it’s true that $$\overrightarrow{a}-\overrightarrow{b}=\overrightarrow{a}+(-\overrightarrow{b})$$.

In order to find the sum of two vectors, $$\overrightarrow{AB}$$ and $$\overrightarrow{BC}$$, if they are collinear and have same direction, we need to add modules of these vectors. $$|\overrightarrow{AB}|+|$$$$\overrightarrow{BC}|=|$$$$\overrightarrow{AC}|$$ (Picture 9).

In order to find the sum of two vectors $$\overrightarrow{AB}$$ and $$\overrightarrow{BC}$$, if they are collinear and have different direction, we need to subtract the module of the smaller vector from the module of the bigger vector. $$|\overrightarrow{AB}|-|$$$$\overrightarrow{BC}|=|$$$$\overrightarrow{AC}|$$(Picture 10).

Multiplication of vectors

Vector $$\overrightarrow{b}$$ is the product of nonzero vector $$\overrightarrow{a}$$ and number k, that is not equal to 0, if $$|\overrightarrow{b}|=|k||\overrightarrow{a}|$$.

If k>0, then $$\overrightarrow{a}↑↑$$$$\overrightarrow{b}$$; if k<0, then $$\overrightarrow{a}↑↓$$$$\overrightarrow{b}$$.
If vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ are collinear and $$\overrightarrow{a}≠$$$$\overrightarrow{0}$$ exist, so number k that
$$\overrightarrow{b}=$$$$k\overrightarrow{a}$$.
If vector $$\overrightarrow{a}$$ has coordinates $$(x_{a};y_{a})$$, then vector k$$\overrightarrow{a}$$ has coordinates
$$(kx_{a};ky_{a})$$.