# what is dot and cross products?

I know how to calculate both of them
and I know I can get the angle between the 2 vectors using them
but is that it ?
what do they even represent ?
like when I get the result of a dot product what should I call the result
what is the result
same with cross product is the resulting vector of a cross product represents something ?

• Just because an operation between two objects gives you an output, doesn't mean that the output is immediately something tangible or useful: however this doesn't mean it isn't useful. Perhaps it can be used in conjunction with something else. So although a . b is just a number which is mostly meaningless, that doesn't mean the dot product doesn't have it's uses. Similarly, in quantum mechanics the wavefunction even has a name, but just by looking at it you don't get anything useful. You have to extract the information from the wavefunction in order to see it's uses. Mar 24, 2022 at 23:34
• The vector cross product, however, does give you a vector perpendicular to two vectors. So the vector product is immediately useful. Mar 24, 2022 at 23:35
• This is quite a broad topic. For the geometric interpretation of the cross product you may want to read this. Mar 24, 2022 at 23:40
• Actually it's two distinct broad topics--one about dot products, one about cross products--three if you count math.stackexchange.com/q/1813988/139123. There are already several questions on this site about either the dot product or cross product; you could see if they answer your questions. Mar 25, 2022 at 0:33
• so I can use the cross product in something like finding the level Equation then maybe thats why when you use cross product on a 2 dementional vectors the result is on the 3rd demention nice thx Mar 25, 2022 at 8:53

The dot product between two vectors $$\underline{u}$$ and $$\underline{v}$$ in Euclidean space $$\mathbb{R}^3$$ is denoted $$\underline{u}\bullet\underline{v}.$$ It acts as a function that takes a pair of vectors $$(\underline{u},\underline{v})$$ as the input, and it returns a real number as the output. What exactly does this real number represent? It represents the projection of $$\underline{u}$$ into the "line" generated by $$\underline{v}.$$ To put it into more intuitive terms, the dot product $$\underline{u}\bullet\underline{v}$$ is a quantitative measure of how close $$\underline{u}$$ and $$\underline{v}$$ are to being parallel vectors. Think about this: if $$\underline{u}$$ and $$\underline{v}$$ are unit vectors, then $$\underline{u}\bullet\underline{v}=-1$$ tells you that $$\underline{u}$$ and $$\underline{v}$$ are anti-parallel, and this makes sense: $$-1$$ is the smallest possible value, and thus it tells you that anti-parallel vectors are the furthest away from being parallel (hence the name anti-parallel). If $$\underline{u}\bullet\underline{v}=0,$$ then that tells you they are not close to being parallel or anti-parallel, the situation is perfectly in the middle. In other words: $$\underline{u}$$ and $$\underline{v}$$ are perpendicular to one another. In mathematics, we say $$\underline{u}$$ and $$\underline{v}$$ are orthogonal. But if $$\underline{u}\bullet\underline{v}=1,$$ the greatest possible value, then that tells you that $$\underline{u}$$ and $$\underline{v}$$ actually are parallel. In fact, they are the same vector.
The cross product between two vectors $$\underline{u}$$ and $$\underline{v}$$ in Euclidean space $$\mathbb{R}^3$$ is denoted $$\underline{u}\times\underline{v}.$$ It acts as a function that takes a pair of vectors $$(\underline{u},\underline{v})$$ as the input, and it returns a third vector $$\underline{w}$$ as the output. What exactly does this vector represent? It is the vector that is perpendicular to the plane spanned by the vectors $$\underline{u}$$ and $$\underline{v}.$$ This is useful, because if $$\underline{u}$$ and $$\underline{v}$$ are themselves mutually perpendicular, then $$\underline{u},$$ $$\underline{v}$$ and $$\underline{w}$$ will all be pairwise perpendicular, and so this forms a basis for $$\mathbb{R}^3,$$ and this basis is orthogonal. This is important because we often care about having the basis resemble something like Cartesian coordinates, which are computationally very simple.