To find the location of point [tex]\( R \)[/tex] on the number line, we'll use the section formula, given that point [tex]\( R \)[/tex] divides the directed line segment from point [tex]\( Q \)[/tex] at [tex]\(-2\)[/tex] to point [tex]\( S \)[/tex] at [tex]\(6\)[/tex] in a [tex]\(3:2\)[/tex] ratio.
The section formula for finding the coordinates of a point that divides a line segment in a given ratio is:
[tex]\[ R = \frac{mx_2 + nx_1}{m + n} \][/tex]
Here,
- [tex]\( m = 3 \)[/tex]
- [tex]\( n = 2 \)[/tex]
- [tex]\( x_1 = -2 \)[/tex] (coordinate of [tex]\( Q \)[/tex])
- [tex]\( x_2 = 6 \)[/tex] (coordinate of [tex]\( S \)[/tex])
Substitute these values into the formula:
[tex]\[ R = \frac{3(6) + 2(-2)}{3 + 2} \][/tex]
Now, perform the multiplications and additions inside the fraction:
[tex]\[ R = \frac{18 + (-4)}{5} \][/tex]
[tex]\[ R = \frac{18 - 4}{5} \][/tex]
[tex]\[ R = \frac{14}{5} \][/tex]
Thus, the location of point [tex]\( R \)[/tex] on the number line is:
[tex]\[ \boxed{\frac{14}{5}} \][/tex]
