Why do we need both dot product and cross product? I was looking for an intuitive definition for dot product and cross product. I have found two similar quesitions in SO, but I am not satisfied with the answers. Finally I found a possible answer here. It says dot product actually gives us a way to depict mathematically how parallel two lines are and on the other side cross products tells us how two lines are perpendicular to each other. So my question is why do we want both. Why cant we just have dot product?
 A: There is a sense where you can start with just a scalar product.
If you assume that $v^2=vv=v\cdot v=|v|^2$ holds for any vector then that is a scalar product for a vector with itself. 
From that you can get the scalar product of any two vectors $v$ and $w$, $v \cdot w= \frac{1}{2}\left((v+w)^2-v^2-w^2\right)=\frac{1}{2}\left(vw+wv\right)$ where for the last equality we assume the same distributive laws as for matrices.
So the simplest assumption of having a scalar product of any vector with itself can easily give us a scalar product of any two vectors (we can use the scalar products of $v+w, v$ and $w$ with themselves).
But we can also have a general product and allow it to have the regular rules of matrix algebras (associative, distributive, multiplication, addition, scaling by scalars, etcetera). Then we note that for vectors $v$ and $w$ we have $vw=\frac{1}{2}\left(vw+wv\right)+\frac{1}{2}\left(vw-wv\right)$ or $vw=v\cdot w+\frac{1}{2}\left(vw-wv\right)$ and we can notice that the last term $\frac{1}{2}\left(vw-wv\right)$ is different than a scalar or a vector, specifically its square is negative. For vectors $v$ and $w$ we can denote $\frac{1}{2}\left(vw-wv\right)$ by $v \wedge w$.
These new objects, like $v\wedge w$ naturally represent the plane spanned by the vectors $v$ and $w$. They are useful in physics, though for historical reasons we often use the complete accident that in 3d there are vectors orthogonal to planes to represent $v \wedge w$ by the vector orthogonal to it and then call it the cross product.
When you have both products $v \cdot w$ and $v\wedge w$ you have the full information to get the product $vw=v \cdot w+v\wedge w$, and from that you can solve for $v$ or $w$ if you have the other one.  For instance $v=vw\frac{w}{w^2}=\left(v \cdot w+v\wedge w\right)\frac{w}{w^2}$. So therefore if you know the scalar product and the wedge product and you know $w$ then you can find out what $v$ was. Neither of the products by themselves allow you that. Many vectors can give the same inner product, and many vectors can give the same wedge product, but knowing both can allow you know the full relative relationship between the two.
It all follows from just the scalar product, but you get the whole package if you want it.

tl;dr
The dot product (symmetric part of the one product) tells you how much they have in common.  The other product (antisymmetric part of the one product) tells you how much they orthogonal, specially how much you have to rotate one line to get align it with the other, and if you don't live in just a plane it also tells you the plane in which you need to rotate to send one into the other. That directionality is something you don't get from just a scalar.
But you don't need two products. One product is enough as long as you do it the invertible way. And that way naturally creates numbers, lines, planes and even higher dimensional objects as they come up. Further details above.
A: The dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. This operation can be defined either algebraically or geometrically.
The cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product a × b of two linearly independent vectors a and b is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming.
As both of them has got different aspects , we have to use both of them.
A: In short:


*

*The dot product gives you the multiplication of the parallel components.

*

*Example: The work expression $W=\vec F \cdot \vec r$, where only the force component parallel to the direction (or likewise, the position component parallel to the force) is wanted.


*The (magnetude of the) cross product gives you the multiplication of the perpendicular components.

*

*Example: The torque expression $\vec \tau=\vec F \times \vec r$, where only the force component perpendicular to the "arm" (or likewise vise versa) is wanted.




Why cant we just have dot product?

Simply because we sometimes want the perpendicular variation and not only the parallel one. At some point the cross product was "invented" to describe this in a neat and concise expression.
Just keep in mind that while the dot product (or scalar product) gives you a number only (a scalar), the cross product (or vector product) gives you a number with a direction (that is, a vector). And this direction is perpendicular to both.
A: The dot and cross products are both recovered as components of the tensor product, which takes two vectors and gives you a rank-2 tensor $v^\mu w^\nu$. The dot product is the trace of this tensor, obtained by a contraction $v^\mu w_\mu$ -- this is useful because it is a scalar, which is invariant under nice transformations (rotations and skews, specifically). The cross product is the anticommutativity of this, evaluated as $v^\mu w^\nu-v^\nu w^\mu$ (for two vectors -- for more vectors you need to use a Le Cevita symbol to sum with signs over the permutations of $\mu,\nu...$).
