Vectors
5.0 Vector Joining Two Points
5.0 Vector Joining Two Points
If two vectors are $A({x_1},{y_1},{z_1})$ and $B({x_2},{y_2},{z_2})$, then the vector joining these two points is given by $\overrightarrow {AB} $.
Join the points $A$ and $B$ with the origin as shown in figure and applying the triangle law of vector addition in triangle $OAB$, we have
$$\overrightarrow {OA} + \overrightarrow {AB} = \overrightarrow {OB} $$
Using vector addition properties, we have $$\overrightarrow {AB} = \overrightarrow {OB} - \overrightarrow {OA} $$
$$\begin{equation} \begin{aligned} \overrightarrow {AB} = ({x_2}\widehat i + {y_2}\widehat j + {z_2}\widehat k) - ({x_1}\widehat i + {y_1}\widehat j + {z_1}\widehat k) \\ \overrightarrow {AB} = ({x_2} - {x_1})\widehat i + ({y_2} - {y_1})\widehat j + ({z_2} - {z_1})\widehat k \\\end{aligned} \end{equation} $$
The magnitude of vector $\overrightarrow {AB} $ is given as $$\sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}} $$
Question 10. Find the vector joining the points $P(2, - 1,3)$ and $Q(3, - 6,9)$ directed from $P$ and $Q$.
Solution: Since the vector is to be directed from $P$ to $Q$, $P$ is the initial point and $Q$ is the terminal point. So, the required vector joining $P$ to $Q$ is the vector $\overrightarrow {PQ} $ i.e.,
$$\begin{equation} \begin{aligned} \overrightarrow {PQ} = (3 - 2)\widehat i + ( - 6 + 1)\widehat j + (9 - 3)\widehat k \\ \overrightarrow {PQ} = \widehat i - 5\widehat j + 6\widehat k \\\end{aligned} \end{equation} $$
