Prove orthogonality with points using orthogonal projection

Question

I need to prove that $M(-1;-1)$ can have an orthogonal projection on the segment $[AB]$, Where $A(-2;4)$ and $B(1;2)$.

I think that I need to prove by the dot product which will be equal to 0, but I can't go further. How can I do that ?

Answer 1

I bet there are multiple ways to solve this exercise. Here's one possible approach.

Let's say the base of that desired orthogonal projection is some point $P$. Since $P$ lies on the segment $AB$, we can say that $\vec{AP}=t\vec{AB}$ for some $0\le t\le1$. So the coordinates of $P(x_P;y_P)$ are given by $x_P=x_A+t(x_B-x_A)$ and similarly for $y_P$.

Also the vector $\vec{MP}$ must be orthogonal to the vector $\vec{AB}$, i.e. their dot product must be $0$, which gives us an equation to solve for $t$.