Vector operations

Vector addition

1. Addition of any two vectors according to the triangle rule.
To add two vectors $\vec{a}$ and $\vec{b}$, we set aside vector $\vec{a}$ from any point on the plane, and vector $\vec{b}$ from the end point of vector $\vec{a}$. Then the sum of vectors is called a vector $\vec{c}$ with the beginning at the initial point of vector $\vec{a}$ and the end point at the end point of the vector $\vec{b}$:
Thus, vector $\vec{c}$ shows the displacement of the end point of vector $\vec{a}$ as a result of adding vector b to it. $\vec{b}$. If we denote vectors by two letters, we obtain the rule for adding two vectors $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{BC}}$ in letter form:

$\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{AC}}$

If we generalize this result to the sum of several vectors, we obtain the polygon rule for adding any number of vectors: the sum of several vectors is a vector with a beginning at the initial point of the first vector and an end at the end point of the last vector: 2. Addition of two non-collinear vectors using the parallelogram rule. If we plot two non-collinear vectors $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{AD}}$ from one point A of the plane and construct a parallelogram ABCD on these vectors, then the sum of the vectors ABCD, $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{AD}}$ will be the vector $\overrightarrow{\mathrm{AC}}$ (the diagonal of the parallelogram), because . Really,
$$\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{AD}}=\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{AC}}$$

Subtraction of vectors

We obtain the rule for subtracting vectors using the definition of vector addition. The difference $\vec{c}$ between vectors $\vec{a}$ and $\vec{b}$ is the sum of the vector $\vec{a}$ and the vector opposite to the vector $\vec{b}$:

$$\vec{c}=\vec{a}-\vec{b}=\vec{a}+(-\vec{b})$$
Thus, to obtain the difference of vectors $\vec{c}=\vec{a}-\vec{b}$, we need to set aside the vectors $\vec{a}$ and $\vec{b}$ from one point on the plane and connect the endpoints of these vectors with a segment directed toward the vector from which the subtraction is performed, that is, toward the vector $\vec{a}$.

Multiplication of a vector by a number

For any number p and any vector $\vec{a}$ the vector p$\vec{a}$ is codirected with the vector $\vec{a}$ for p≥0, and in the opposite direction for p<0, and the length of the vector p$\vec{a}$ is equal to |p|⋅|$\vec{a}$|.
From the definition of multiplication of a vector by a number it follows that

0⋅$\vec{a}$ = $\vec{0}$;    p⋅$\vec{0}$ = $\vec{0}$.

Properties of vector operations

For any vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ and for arbitrary numbers p and q the equalities are valid:
1) Commutativity property: $\vec{a}+\vec{b}=\vec{b}+\vec{a}$,   $\vec{a}\cdot \; p\; =p\; \cdot \vec{a}$;
2) Associativity property: $\vec{a}+\; (\vec{b}+\vec{c})\; =\; (\vec{a}+\vec{b})\; +\vec{c}$;
3) The property of a neutral element by addition: $\vec{a}+\vec{0}=\vec{a}$;
4) $\vec{a}+\; (\; -\vec{a})\; =\vec{0}$;
5) $\vec{a}\cdot \; 1\; =\vec{a}$;
6) The associative property of the operation of multiplying a vector by a number: $(p\; \cdot \; q)\; \cdot \vec{a}=\; p\; \cdot \; (q\; \cdot \vec{a}$);
7) Distributive properties of multiplication of a vector by a number: $p\; \cdot \; (\vec{a}+\vec{b})\; =\; p\; \cdot \vec{a}+\; p\; \cdot \vec{b}$,   $(p\; +\; q)\; \cdot \vec{a}=\; p\; \cdot \vec{a}+\; q\; \cdot \vec{a}$;
The properties of commutativity and associativity allow vectors to be added in any order.