How to Find the Vector Between Two Points

A vector PQ from point P to point Q

The vector between two points can easily be expressed using the position vectors of each point. The point $P = (x_{1}, y_{1})$ can be rewritten as a position vector: $\vec{O P} = (x_{1}, y_{1})$ . You do the same with the other point $Q = (x_{2}, y_{2})$ , and get $\vec{O Q} = (x_{2}, y_{2})$ . You can now create the vector between the two points by subtracting the end point from the starting point, which is the difference $\vec{O Q} - \vec{O P}$ . This is how you find the vector between two points:

Rule

The Vector Between Two Points

\begin{array}{l} \vec{P Q} & = \vec{O Q} - \vec{O P} \\ = (x_{2}, y_{2}) - (x_{1}, y_{1}) \\ = (x_{2} - x_{1}, y_{2} - y_{1}) \end{array}

Example 1

You have the points $A = (2, 6)$ and $B = (- 3, - 7)$ . Find $\vec{A B}$ .

Rewrite the points as position vectors:

\vec{O A} = (2, 6), \vec{O B} = (- 3, - 7) .

Next, find $\vec{A B}$ :

\begin{array}{l} \vec{A B} = \vec{O B} - \vec{O A} & = (- 3, - 7) - (2, 6) \\ = (- 5, - 13) . \end{array}

\vec{A B} = \vec{O B} - \vec{O A} = (- 3, - 7) - (2, 6) = (- 5, - 13) .

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