The vector v has initial point P and terminal point Q. How do you write v in the form $ai + bj$, that is, find its position vector given $P = \left( {5,4} \right)$; $Q = \left( {7,8} \right)$

Hint: Given the points. We have to find the vector equation in the form $ai + bj$. First, we will find the position vector v by subtracting the x-values and y-values of two points P and Q. Then, represent the coordinates in the form $ai + bj$.
Formula used:
The position vector is given by:
$v = PQ = \left( {{x_2} - {x_1},{y_2} - {y_1}} \right)$
Where $\left( {{x_1},{y_1}} \right)$represent the initial point and $\left( {{x_2},{y_2}} \right)$ represent the terminal point.

Complete step by step solution:
We are given the coordinates $P = \left( {5,4} \right)$and $Q = \left( {7,8} \right)$
Substitute the points $\left( {{x_1},{y_1}} \right) = \left( {5,4} \right)$ and $\left( {{x_2},{y_2}} \right) = \left( {7,8} \right)$ in the equation $v = PQ = \left( {{x_2} - {x_1},{y_2} - {y_1}} \right)$ to find the position vector.
$ \Rightarrow v = \left( {7 - 5,8 - 4} \right)$
Simplifying the expression, we get:
$ \Rightarrow v = \left( {2,4} \right)$
Now, we will write the coordinates of the position vector in the form $ai + bj$
$ \Rightarrow v = 2i + 4j$
Hence, the position vector is $v = 2i + 4j$

Additional information: A vector is always a starting point and the terminal point. The position vector is then determined by subtracting the starting point from terminal point. A vector is represented in magnitude and direction form. Therefore, a vector is a kind of line segment with direction. This position vector is the ordered pair which is used to define the changes in the x and y-values. Two vectors are known as equal vectors if they have the same magnitude and direction, otherwise if the direction is the same or opposite the vectors are known as parallel.

Note: In such types of questions, the students must remember that we subtract the magnitude of the two vectors with the same direction.
Now, the position vector we have found is $v = 2i + 4j$, therefore the x-coordinate of the resultant vector is 2 and the y-coordinate of the vector is 4. The students must also know that a magnitude of a particular vector can be represented as $\sqrt {{a^2} + {b^2}} $.