what is dot and cross products?

Question

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 ?

Answer 1

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.