Vector coordinates

Decomposition of a vector into coordinate vectors

Let's consider a rectangular coordinate system XOY. From the origin O we set aside vectors of unit length $\vec{i}$ and $\vec{j}$ such that the direction of the vector $\vec{i}$ coincides with the direction of the OX axis, and the direction of the vector $\vec{j}$ coincides with the direction of the OY axis. These vectors are called coordinate vectors.
Since vectors $\vec{i}$ и $\vec{j}$ are not collinear, any vector $\vec{a}$ can be represented as:

$\vec{a}=\; x\; \cdot \vec{i}+\; y\; \cdot \vec{j}$.

The coefficients x and y are called the coordinates of the vector $\vec{a}$ in this coordinate system: $\vec{a}\mathrm{(x;\; y)}$ или $\vec{a}\mathrm{\{x;\; y\}}$. The zero vector and coordinate vectors are denoted by $\vec{0}\mathrm{(0;\; 0)}$, $\vec{i}\mathrm{(1;\; 0)}$ and $\vec{j}\mathrm{(0;\; 1)}$, respectively, or $\vec{0}\mathrm{\{0;\; 0\}}$, $\vec{i}\mathrm{\{1;\; 0\}}$ и $\vec{j}\mathrm{\{0;\; 1\}}$.
The decomposition of the zero vector can be represented as follows:

$\vec{0}=\; 0\; \cdot \vec{i}+\; 0\; \cdot \vec{j}$.

The coefficients of equal vectors are correspondingly equal. Conversely, if the coefficients of vectors are correspondingly equal, then the vectors themselves are also equal.

The length of a vector or its modulus in coordinates

Let the coordinates of the vector be given $\vec{a}${x; y}, that is, the initial point of the vector $\vec{a}$ has coordinates (0;0), and the end point - (x;y). Then, using the formula for the length of a segment, we obtain a formula for determining the modulus of a vector:

Coordinates of the sum and difference of vectors

Let the vectors be given $\vec{a}${x₁,y₁} and $\vec{b}${x₂,y₂}: Let's find the coordinates {x,y} of the vector $\vec{c}=\vec{a}+\vec{b}$, используя геометрическое определение суммы векторов:
Since AC=OB and DF=OE, then x = x₁ + x₂, y = y₁ + y₂. Thus, the coordinates of the sum of two vectors are equal to the sum of the corresponding coordinates of these vectors.
You can derive these formulas using the properties of vector addition and vector multiplication by a number: Obviously, to find the coordinates of the sum of several vectors, you need to add up the corresponding coordinates of these vectors.
In a similar way, formulas for the coordinates of the difference of two vectors are obtained $\vec{d}=\vec{a}-\vec{b}$. The coordinates of the difference of two vectors are equal to the difference of the corresponding coordinates of these vectors.

Coordinates of the vector, if the coordinates of its start and end points are given

Let the coordinates of the initial point A(x₁, y₁) of the vector $\overrightarrow{\mathrm{AB}}$ and the end point B be given B(x₂, y₂).
Since $\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}$,then the vector $\overrightarrow{\mathrm{AB}}$ has coordinates

{x₂ - x₁, y₂ - y₁}.

Thus, to find the coordinates of the vector $\overrightarrow{\mathrm{AB}}$, we must subtract the corresponding coordinates of the initial point from the coordinates of the end point of the vector, and the formula for calculating its modulus is:

Coordinates of the product of a vector by a number

The coordinates of the product of a vector $\vec{a}${x, y} by a number q are equal to the products of the corresponding coordinates by this number: Therefore, the coordinates of the vector q⋅$\vec{a}${qx, qy}.