I know how to perform a dot product but don’t really ‘get’ what it means, like how I ‘get’ that length times width gives me a rectangular area. Help me understand intuitively what the dot product operation actually represents.
I will attempt to explain it to you in the way it makes sense to me.
First of all, finding the dot product of two vectors, A and B, does not give you rectangular area. That only happens when you do the cross product on them and even then it would be the area of a parallelogram. If you remember from geometry, not all parallelograms are rectangles but all rectangles are parallelograms, so I would not make that statement that has ambiguity. Cross product gives you the area of parallelogram, not area of rectangle.
Now to answer your question:
When we take the dot product of two vectors, we just want to know what portion of one vector is in the direction of the other.
Suppose you have two Vectors A and B in 2D space and the angle between them is 30 degrees. Intuitively, If we take a flashlight to the top of one vector, the shadow casted on the other would be what we are looking for.
Now mathematically, we describe it as the portion of the vector on the top in the same direction as the vector of the bottom and we give it is own little formula.