I am trying to understand linear algebra for some data science self study that I am doing. One of the basic ideas is the vector dot product. Now I have seen videos explaining that tell the geometric intuition of a dot product of $\vec{v}.\vec{w}$ is that when $\vec{v}$ is projected onto ${\vec{w}}$.
How that is done is by dropping a perpendicular from the edge of $\vec{w}$.
But my question is how is that geometrically possible when $\vec{v}$ is just in the exact opposite direction of $\vec{w}$ ?
For example:
How would you get a perpendicular from the end of $\vec{v}$ to the end of $\vec{w}$ or vice versa. In fact you could come up with a variety of configuration where the perpendicular from the end of 1 vector would never meet the second vector.
So how does vector dot product work geometrically in these cases ?
Imagine a series of vectors converging toward one of the vector you draw and draw for each of them the perpendicular projection along the axis led by the other vector. You'll figure out that their perpendicular projection gets closer and closer from the actual vector.
So to answer your question, in the case the vectors are collinear (along the same axis), their projection is "just themselves", don't forget to add a minus sign to their norms while doing the dot product in the case they are pointing in an opposite direction.
Hope it helps and that I'm clear enough, I'm not an English native so it's sometimes difficult for me to be as clear as I'd like to be.