A point C between two other points A and B, in the line joining A to B, is a linear combination of A and B.
If C is a/n the distance from B to A, then it is (n-a)/n the distance from A to B. We can write that as:
[tex]C=\frac{a}{n}A+\frac{n-a}{n}B[/tex]Notice that the farther it is from B (as a increases), the larger is the factor (a/n) by which we multiply A.
Step 1
Identify the points A and B, and the fractions a/n and (n-a)/n.
We have:
[tex]\begin{gathered} A=(3,-4) \\ B=(-4,4) \\ \\ \frac{a}{n}=\frac{3}{5} \\ \\ \frac{n-a}{n}=\frac{5-3}{5}=\frac{2}{5} \end{gathered}[/tex]Step 2
Use the previous result in the formula to find C:
[tex]\begin{gathered} C=\frac{3}{5}(3,-4)+\frac{2}{5}(-4,4) \\ \\ C=\mleft(\frac{9}{5},-\frac{12}{5}\mright)+\mleft(\frac{-8}{5},\frac{8}{5}\mright) \\ \\ C=\mleft(\frac{9-8}{5},\frac{-12+8}{5}\mright) \\ \\ C=\mleft(\frac{1}{5},-\frac{4}{5}\mright) \end{gathered}[/tex]Answer
Therefore, the point that is 3/5 the distance from B to A is
[tex]\mleft(\frac{1}{5},-\frac{4}{5}\mright)[/tex]
