Dot and cross product

Question

I am unable to understand that why do we have two ways to multiply two vectors? (viz. Dot product and cross product) How do we even know when to use which product? Is it analogous to anything with numbers? Do we multiply numbers in two ways? Or is this concept of multiplication really just another unrelated independent operation unfortunately named multiplication due to lack of alphabet, maybe?

Basically, my question is why do we have two ways for multiplying vectors and how whoever thought of it, thought that we needed two ways?

Answer 1

We use dot product to find out angle between vectors and it's define Euclidean space, which is also normed space with norm $||x|| = \sqrt{(x, x)}$.

Vector product is used to calculate the parallelogram area, which is built on this vectors, and in result we get also vector. Vector product defines Lie algebra with multiplication operation.

These are different mathematical operations that result in different mathematical entities.

Answer 2

The dot product gives a number as output, and it's an operation that comes up naturally when you study the angle between two vectors: the angle $\theta$ between vectors $\mathbf{v}$ and $\mathbf{u}$ is given by $\displaystyle \cos \theta = \frac{\mathbf{v}\cdot \mathbf{u}}{\|\mathbf{v}\|\|\mathbf{u}\| }$. This product makes sense in any number of dimensions. So for example, in 4 dimensions, we could still define the angle between two vectors such as $(2,-1,3,6)$ and $(4,0,-3,7)$, even though we cannot visualize them...

The cross product on the other hand is only defined in three dimensions, and gives a vector as answer. So it is quite a different kind of thing. It comes up when you want to go from 2 to 3 dimensions in the following way: if you start with two 2-dimensional vectors such as $\mathbf{v}=(3,4)$ and $\mathbf{u}=(-1,3)$, we can think of them as actually living in 3-space by adding a $0$ as third coordinate: $\mathbf{v}=(3,4,0)$ and $\mathbf{u}=(-1,3,0)$. Then if we take the cross product $\mathbf{v}\times \mathbf{u}$, the result will be a vector perpendicular to both of them, so it will point in the third dimension, the $z$-axis...

Answer 3

They are both called product / multiplication not because we couldn't think of other names, but rather because they share some very handy properties (like distribution), and giving them related names emphasises this.

And they both have their uses. Scalar product is used when parallel vectors work together and orthogonal vectors cancel, such as with applying a force to a moving object and asking about the work done. Cross product is mostly used when orthogonal vectors work together and parallel vectors cancel, such as when figuring out the area of a parallelogram spanned by two vectors.

It's not some conspiracy to force you to remember two different things. They are genuinely different with their own separate uses. Thinking about the above paragraph concerning parallel versus orthogonal when you see either of them appear may make it easier for you to understand why one is used over the other.