Question

Why is it necessary to use the absolute value of the difference when finding the distance on a number line, but not necessary when finding the differences of the coordinates in the coordinate plane?"

Answer 1

Remember that the distance on a number line is given by the formula: $d=|x_{2}-x_{1}|$
where 
$d$ is the distance between the points. 
$x_{2}$ is the second point in the number line.  
$x_{1}$ is the first point in the number line.
Now, suppose we are trying to find the distance between the points -1 and -5; our first point is -1, so $x_{1}=-1$, and our second point is $-5$, so $x_{2}=-5$. Suppose we are going to find the distance between the two point without using absolute value:
$d=x_{2}-x_{1}$
$d=-5-(-1)$
$d=-5+1$
$d=-4$
Look what we have here, a negative distance! Since distances cannot be negative, we must use absolute value to always get postie distances between two point on a umber line:
$d=|x_{2}-x_{1}|$
$d=|-5-(-1)|$
$d=|-5+1|$
$d=|-4|$
$d=4$ 

Now, the distance between two coordinates on a plane is given by the formula: $d= \sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}$
$where$
$d$ is the distance between the two coordinates
$(x_{1},y_{2})$ are the coordinates of the first point
$(x_{2},y_{2})$ are the coordinates of the second point 
Notice that $(x_{2}-x_{1})^2$ and $(y_{2}-y_{1})^2$ are squared, so it doesn't matter if we get a negative distance because a negative number raised to an even power (like 2) is always positive; therefore we don't need absolute value in this case because we won't ever get a negative distance.