Coordinates of a point, components of a vector and midpoint of a segment

Coordinates of a point on the plane

Let's see how vectors are used to assign coordinates to the points in the plane.

We consider a fixed point in the plane $O$ (known as origin), and a basis $B=\{\overrightarrow{u},\overrightarrow{v}\}$ of $V_2$ (Space vector of dimension $2$).

Let's remember that a basis of $V_2$ are two linearly independent vectors. The set formed by $O$ and $B=\{\overrightarrow{u},\overrightarrow{v}\}$ constitutes a reference system in the plane, since it allows us to determine the position of any other points on the plane.

This is because any other points $P$ on the plane determines along with point $O$ a vector $\overrightarrow{OP}$. Let $(p_1,p_2)$ be the components of the vector in basis $B$. Then $(p_1,p_2)$ are the coordinates of point $P$ in the reference system $R=\{O;\overrightarrow{u},\overrightarrow{v}\}$ and we write $P=(p_1,p_2)$.

The procedure to find the coordinates of point $P$ in a given reference system is the following :

1. From the points $O$ and $P$ we determine the vector $\overrightarrow{OP}$

2. We express the vector $\overrightarrow{OP}$ as a linear combination of the vectors of the basis$B=\{\overrightarrow{u},\overrightarrow{v}\}$, that is to say, $\overrightarrow{OP}=p_1\cdot \overrightarrow{u}+p_2\cdot \overrightarrow{v}$

3. $P=(p_1,p_2)$

From now on we will consider, as reference system $R$, the one formed by the origin of coordinates $O=(0,0)$ and the canonical basis of $V_2$ $B=\{\overrightarrow{i},\overrightarrow{j}\}$.

Components of a vector determined by two points

Let's see now the way to determine the components of a vector if we know the coordinates of its endpoints:

Let $P=(p_1,p_2)$ and $Q=(q_1,q_2)$ be two points of the plane, and $\overrightarrow{PQ}$ the vector that goes from $P$ to $Q$. Then the components of the vector $\overrightarrow{PQ}$ are $\overrightarrow{PQ}=(q_1-p_1,q_2-p_2)$.

Applying a vector to a point

Given a point $P$ and a vector $\overrightarrow{v}$, the result of applying the vector to the point is a new point $Q$ placed in the direction of $\overrightarrow{v}$ and at a distance $|\overrightarrow{v}|$. (module of the vector $\overrightarrow{v}$)

The coordinates of this new point $Q$ are calculated from those of $P=(p_1,p_2)$ and $\overrightarrow{v}=(v_1,v_2)$ thus $$Q=P+\overrightarrow{v}=(p_1+v_1,p_2+v_2)$$

NOTE: It is very important to bear in mind that this addition operation only makes sense between a point and a vector. We must never add two points, and the result of adding two vectors is another vector and not a point!

Midpoint of a segment

Let's consider now a segment with endpoints $A=(a_1,a_2)$ and $B=(b_1,b_2)$. Let $M=(m_1,m_2)$ be the midpoint of the above mentioned segment. Obviously the above mentioned point satisfies that $\overrightarrow{AB}=2\cdot \overrightarrow{AM}$, or that $(b_1-a_1,b_2-a_2)=2\cdot (m_1-a_1,m_2-a_2)$

Separating component by component we obtain: $$\begin{array}{rcl}{b}_1-a_1& =& 2\cdot (m_1-a_1)\\ b_2-a_2& =& 2\cdot (m_2-a_2)\end{array}$$ and isolating we have: $$\begin{array}{rcl}{m}_1& =& {\displaystyle \frac{a_1+b_1}{2}}\\ m_2& =& {\displaystyle \frac{a_2+b_2}{2}}\end{array}$$ So that we can calculate the coordinates of the midpoint of a segment from the coordinates of its endpoints.