Dot product is a projection, a shadow of one upon the other. If you have perpendicular vectors, no shadow is projected. If you have parallel vectors, you have the maximum. The result is a scalar, proportional to how much of the vector is covered by the others shadow. Cross product is the "area of the parallelogram" made between the vectors. Maxing out when pi/2. To get the right hand rule, remember that a plane can be defined by a normal vector, so which vector defines this area between these to vectors? One that is perpendicular to both. The result is a vector, proportional to that 'area'. Might not make mathematical (nor physical sense), but that's how I remember it. More on reddit.com

Imagine driving a boat in moving water. If the water is going your way then combining Water + Boat gives you more. If the water is against you, then combining probably gives you less. Dot Product is like combining Water + Boat and getting the final distance traveled. Bonus: Now imagine if the boat was going at an angle. Different angles will get you more change. More on reddit.com

Well, I would say that dot products might be somewhat redundant. Inner products, however, aren't. In general vector spaces, you don't have a preexistent notion of distance nor one of angles. Inner products provide a useful way of defining these. Because of this, there are several properties that inner product spaces have and more general vector spaces don't. Another way of looking at this is by saying that the "geometric" vector spaces that you've studied have too much structure (as a side note, these vector spaces are not quite geometric, since the 0 vector is a special point, and geometric spaces don't make differences between their points), and there are situations where what you want are more general spaces that keep some of their properties. The axioms of vector spaces capture a lot of these properties, namely, those related with scaling and addition. Dot products captures the ideas of angle and distance. You could also want distance without angle, and then you study normed vector spaces, and there are more possibilities. So, to answer your question, what dot products do is that they capture certain features of, let's say ℝn, without assuming too much. But this only becomes noticeable when you go deeper into the study of linear algebra. If you restrict yourself to study real coordinate spaces with the usual geometric properties, then yes, dot products are pretty whimsical operations. More on reddit.com

We are going to ELIKnowWhatAVectorIs

Let's say you have two unit vectors (thought of as arrows with length 1 coming from the origin). The dot product is a measurement of "how parallel/perpendicular are these vectors?"

If u dot v = 1, then the unit vectors are exactly parallel. (Try dotting <1,0> with <1,0>)

If u dot v = 0, then the unit vectors are exactly perpendicular (like <1,0> with <0,1>)

If u dot v = -1, then the unit vectors are antiparallel, parallel but in opposite directions (like <1,0> with <-1,0>)

If the vectors aren't unit length, like most vectors, then you have to multiply by the product of the lengths. The larger the dot product (compared to the product of the lengths), the closer the vectors are to parallel, or antiparallel.

For example, if you have a vector whose length is 3, and another vector whose length is 7, and their dot product is -21, then these vectors must be antiparallel.

Here's another case: If you have a vector of length 5 and a vector of length 9, and their dot product is 2, then the vectors are close to perpendicular, because 2 is a very small dot product compared to 45.

Let me know if this helps.

Let me add: generally, for various reasons, it is more useful to know when two vectors are perpendicular, which is good, because solving when an expression is equal to 0 is much easier than trying to solve for the product of the lengths if we don't know the lengths of the vectors. (Example: if I say the dot product of two vectors is 3, you don't know if that is a lot, or if it is very small, compared to the product of the lengths. But if I say the dot product is 0, you know they are perpendicular, no question.) Sometimes you will see statements "like ax+by+cz = 0, therefore such-and-such vectors are perpendicular"; in these cases, they are using a dot product.

More on reddit.com