# Scalar times Point + Scalar times Point?

Let $P$, $Q$ be a pair of points in the Euclidean plane and let $t_1$, $t_2$ be a pair of scalars. My textbook says that the following operations are nonsense:

$$P + Q\\ t_1 \cdot P$$

However $t_1 \cdot P + t_2 \cdot Q$ is completely valid and makes sense!

I cannot understand this at all. If you cannot multiply a scalar by a point and you cannot add points together, then how come that this expression is still valid? And how come that the result is a point?

$t_1P+t_2Q$ only makes sense if $t_1+t_2=1$. The important thing here is that the operation is independent of the choice of coordinate frame. If $t_1+t_2=1$, the point is just an interpolation on the line between the two points. If not, you have a leftover term that shifts the point around depending on where your coordinate origin is.
An interpolation is ok, because you are essentially taking $D=Q-P$, the displacement vector from P to Q. Now start at $P$ and go a fraction of this distance:
$$R=P+t_1(Q-P)=(1-t_1)P+t_1 Q$$
Notice how the sum of the coefficients is $1$.
Formally, you can imagine adding a third component (use homogeneous coordinates). A point has a form $(x,y,1)$ and a displacement is $(x,y,0)$. If your operation sets the third component to $1$, the result is a point, if it is $0$, the result is a vector, and if it's anything else, the result is neither of those things.