# Find the scalar components and magnitude of the vector joining the points $P (x_1, y_1, z_1)$ and $Q (x_2, y_2, z_2)$.

$\begin{array}{1 1} \text{(A) The scalar components of PQ are } (x_2-x_1),\:( y_2-y_1)\;and \;( z_2-z_1),\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} \\ \text{(B) The scalar components of PQ are} (x_1-x_2),\:( y_1-y_2) \;and \;( z_1-z_2),(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2 \\\text{(C) The scalar components of PQ are} x_1\: y_1\; and \;z_1,(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2 \\ \text{(D) The scalar components of PQ are} x_2\: y_2\; and \;z_2,(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2 \end{array}$

Toolbox:
• $\overrightarrow{a} = x_1 \hat{i} + y_1 \hat{j} + z_1 \hat{k}$
• The magnitude of a vector $|\overrightarrow{a}| = \sqrt{x_1^2 + y_1^2 + z_1^2}$
STEP : 1
Let the position vectors of $P (x_1, y_1,z_1)$ and $Q(x_2, y_2, z_2)$ be $\overrightarrow{OP}$ and $\overrightarrow{OQ}$
$\overrightarrow{OP} = x_1 \hat{i} + y_1 \hat{j} + z_1 \hat{k}$
$\overrightarrow{OQ} = x_2 \hat{i} + y_2 \hat{j} + z_2 \hat{k}$
\begin{align*} Therefore \overrightarrow{OP} & = \overrightarrow{OQ} - \overrightarrow{OP} \\ & = (x_2 \hat{i} + y_2 \hat{j} + z_2 \hat{k}) - (x_1 \hat{i} + y_1 \hat{j} + z_1 \hat{k}) \\ & = (x_2 - x_1) \hat{i} + (y_2 - y_1) \hat{j} + (z_2 - z_1) \hat{k} \end{align*}
STEP : 2
The scalar components of vector $\overrightarrow{PQ}$ are $(x_2 - x_1), (y_2 - y_1)$ and $(z_2 - z_1)$
(i.e) $|\overrightarrow{PQ}| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2 - z_1)^2}$