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 \(|\overrightarrow{AB}|=\)\(\sqrt{x^{2}_{a}+y^{2}_{a}}\).

Adding of vectors

      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
\((kx_{a};ky_{a})\).