I have two questions here and I am not sure why the method I used for the first one won't work on the second one.
The first question says the following:
Let $A=(-9,2)$ and $B=(1,-6)$ be points in $\mathbb{R}^2$. Find the point $P$ that is $\frac{1}{3}$ of the way from the point $A$ to the point $B$.
So this means that we have $AP=\frac{1}{3}AB$.
Let the coordinates of $P$ be $(x,y)$. So that means that we have:
$(P-A)=\frac{1}{3}(B-A)$
$(x+9,y-2)=\frac{1}{3}(10,-8)$
$(x+9,y-2)=\left(\frac{10}{3},-\frac{8}{3}\right)$
So that means $x+9=\frac{10}{3}$ and $y-2=-\frac{8}{3}$ or in other words, $x=-\frac{17}{3}$ and $y=-\frac{2}{3}$.
This WAS the correct answer to the first question.
The second question says the following:
Let $A=(0,1)$ and $B=(6,4)$ be points in $\mathbb{R}^2$. Find the point $C$ on the line segment connecting A and B which is $\frac{1}{3}$ of the way from the point $A$ to the point $B$.
The professor solved it using the following method
This has to be a typo right? The professor only seemed to find the distance from $A$ to $C$. He didn't actually find the coordinates of point $C$ it seems.
Is my understanding correct?