Difference between dot product and cross product

Question

It is known that a dot product is a scalar, but a cross product is a vector.

Dot product: $$ A \cdot B \in \mathbb{R} $$

$$ A \cdot B = |A||B| \cos\theta $$

$$ A \cdot B = A_xB_x + A_yB_y + A_zB_z $$

Cross product: $$ A \times B \in \mathbb{R}^3 $$

$$ \| A \times B \| = |A||B|\sin\theta $$

$$ A \times B = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \ A_x & A_y & A_z \\ \ B_x & B_y & B_z \\ \end{vmatrix} $$

$$ A \times B = (A_yB_z-A_zB_y)\mathbf{i} + (A_zB_x - A_xB_z)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k} $$

It is known that the dot product results in a scalar, and the cross product results in a vector. Why does one have direction, and the other does not, and why are they calculated differently?

Answer 1

You a;lready know one produces a scalar, and the other a vector.

Dot product is defined between any two vectors of same (but arbitrary) sizes. However cross-product is defined only for 3D-vectors.

Your definition is presented in a way that expects one to know the angle between the vectors to calculate the dot product. Actually it is the other way: one can calculate the dot product given all the components of the vector, and use that product value to determine the angle between those vectors.

In a theoretical view point dot product makes available a natural isomorphism to the dual vector space. And cross-product gives a map from the second exterior power of the 3-dimensional vectors space to itself, which is an isomorphism. These are the theoretical advantages.

Answer 2

The only thing I think they have in common is that they take two vectors as input and have the word "product" in their name. They don't work in the same dimensions, produce the same types of values or have a meaningful interpretation in terms of the other.