# Find the scalar components and magnitude of the vector joining the points

Question:

Find the scalar components and magnitude of the vector joining the points

$\mathrm{P}\left(x_{1}, y_{1}, z_{1}\right)$ and $\mathrm{Q}\left(x_{2}, y_{2}, z_{2}\right)$.

Solution:

The vector joining the points $\mathrm{P}\left(x_{1}, y_{1}, z_{1}\right)$ and $\mathrm{Q}\left(x_{2}, y_{2}, z_{2}\right)$ can be obtained by,

$\overrightarrow{\mathrm{PQ}}=$ Position vector of $\mathrm{Q}-$ Position vector of $\mathrm{P}$

$=\left(x_{2}-x_{1}\right) \hat{i}+\left(y_{2}-y_{1}\right) \hat{j}+\left(z_{2}-z_{1}\right) \hat{k}$

$|\overrightarrow{\mathrm{PQ}}|=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$

Hence, the scalar components and the magnitude of the vector joining the given points are respectively $\left\{\left(x_{2}-x_{1}\right),\left(y_{2}-y_{1}\right),\left(z_{2}-z_{1}\right)\right\}$ and $\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$.