The visual geometric interpretation of a “Dot Product” is to take two vectors rooted at the origin, V and W, and project W onto the same path as V. The Dot Product is the same as then taking the length of W multiplied by the length of V. If W is instead pointing in the opposite direction of V, but still on the same line, that Dot Product would actually be negative.
This is done mathematically by multiplying each row of the two vectors and then adding the products together.
Connections
Multiplying Two Matrices Is The Math Behind Applying Two Transformations To A Vector
Link Explanation: The linked note discusses how the multiplication of two matrices can be represented by applying two linear transformations to a vector. This is similar but also quite different than a dot product and it should not be confused between the two. The dot product is the result of projecting one vector onto the infinite straight line path of another vector. The distinction is that there is no change to the grid, or more accurately, to i-hat and j-hat.
Reference
https://www.youtube.com/watch?v=LyGKycYT2v0&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=10