Vector Joining Two Points

If ${\displaystyle P_1(x_1,y_1,z_1)}$ and ${\displaystyle P_2(x_2,y_2,z_2)}$ are any two points, then the vector joining ${\displaystyle P_1}$ and ${\displaystyle P_2}$ is the vector  in following fig.

Joining the points ${\displaystyle P_1}$ and ${\displaystyle P_2}$ with the origin O, and applying triangle law, from the triangle ${\displaystyle O{P}_1{P}_2}$, we have
 ${\displaystyle \overrightarrow{O{P}_1}+\overrightarrow{P_1{P}_2}=\overrightarrow{O{P}_2}}$
Using the properties of vector addition, the above equation becomes
${\displaystyle \overrightarrow{P_1{P}_2}=\overrightarrow{O{P}_2}-\overrightarrow{O{P}_1}}$
i.e. ${\displaystyle \overrightarrow{P_1{P}_2}=(x_2\hat{i}+y_2\hat{j}+z_2\hat{k})-(x_1\hat{i}+y_1\hat{j}+z_1\hat{k})}$
= ${\displaystyle (x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}}$

The magnitude of vector ${\displaystyle \overrightarrow{P_1{P}_2}}$ is given by
${\displaystyle \left|\overrightarrow{P_1{P}_2}\right|=\sqrt{{(x_2-x_1)}^2 +{(y_2-y_1)}^2+{(z_2-z_1)}^2}}$
 Video link : https://youtu.be/2zP78yNYbMs