If addition of two vectors $\vec{v}$ and $\vec{w}$ is given by the equation $(v_x+w_x, v_y+w_y,v_z+w_z)$, you might think that the product of two vectors is $(v_xw_x, v_yw_y,v_zw_z)$, but you would be wrong. This way of multiplying vectors is not used in practice. We will define two other useful ways to multiply vectors in this section.
The dot product tells you how similar two vectors are to each other: \[ \vec{v}\cdot\vec{w}\equiv v_xw_x+v_yw_y+v_zw_z \equiv \|\vec{v}\|\|\vec{w}\|\cos(\varphi) \quad \in \mathbb{R}, \] where $\varphi$ is the angle between the two vectors. The factor $\cos(\varphi)$ is largest when the two vectors point in the same direction.
The formula for the cross product is more complicated so I will not show it to you just yet. What is important is that the cross product of two vectors is another vector: \[ \vec{v}\times\vec{w} = \{ \text{ a vector perpendicular to both } \vec{v} \text{ and } \vec{w} \ \} \quad \in \mathbb{R}^3. \] If you take the $\times$ product of one vector that points in the $x$ direction with another vector in the $y$ direction, you will get a vector in the $z$ direction.
The dot product between two vectors is given by the formula: \[ \vec{v}\cdot\vec{w}\equiv v_xw_x+v_yw_y+v_zw_z \equiv \|\vec{v}\|\|\vec{w}\|\cos(\varphi) \in \mathbb{R}, \] where $\varphi$ is the angle between the two vectors. This operation is also known as the inner product or scalar product. The name scalar comes from the fact that the result of the dot product is a scalar number: a number that does not change when the basis changes.
The signature for the dot product operation is \[ \cdot : \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}. \] The dot product takes two vectors as inputs and outputs a real number.
The geometric factor $\cos(\varphi)$ depends on the relative orientation of the two vectors:
$\cos(\varphi)=\cos(0^\circ) = 1$ and so
$\vec{v}\cdot\vec{w}=\|\vec{v}\|\|\vec{w}\|$.
* If the vectors are perpendicular to each other,
$\cos(\varphi)=\cos(90^\circ) = 0$ and so
$\vec{v}\cdot\vec{w}=\|\vec{v}\|\|\vec{w}\|(0)=0$.
* If the vectors point in exactly opposite directions, then
$\cos(\varphi)=\cos(180^\circ) = -1$ and so
$\vec{v}\cdot\vec{w}=-\|\vec{v}\|\|\vec{w}\|$.
The cross product takes as inputs two vectors and returns another vector: \[ \times : \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3. \] The fact that the output of this operation is a vector is why we sometimes refer to the cross product as the vector product.
The cross products of the individual basis elements is defined as follows: \[ \hat{\imath}\times\hat{\jmath} =\hat{k}, \ \ \ \hat{\jmath}\times\hat{k} =\hat{\imath}, \ \ \ \hat{k}\times \hat{\imath}= \hat{\jmath}. \]
The cross product is anti-symmetric in its inputs, which means that swapping the order of the inputs introduces a negative sign in the output: \[ \hat{\jmath}\times \hat{\imath} =-\hat{k}, \ \ \ \hat{k}\times\hat{\jmath} =-\hat{\imath}, \ \ \ \hat{\imath}\times \hat{k} = -\hat{\jmath}. \] I bet you had not seen an anti-symmetric product before. Most products you have seen so far in math are commutative, which means that the order of the inputs doesn't matter. The product of two numbers is commutative $ab=ba$, the dot product is commutative $\vec{u}\cdot\vec{v}=\vec{v}\cdot\vec{u}$, but the cross product of two vectors is non commutative $\hat{\imath}\times \hat{\jmath} \neq \hat{\jmath}\times \hat{\imath}$.
For two arbitrary vectors $\vec{a}=(a_x,a_y,a_z)$ and $\vec{b}=(b_x,b_y,b_z)$, the cross product is calculated as follows: \[ \vec{a}\times\vec{b}=\left( a_yb_z-a_zb_y, \ a_zb_x-a_xb_z, \ a_xb_y-a_yb_x \right). \]
The length of the output of the cross product is proportional to the $\sin$ of the angle between the vectors: \[ \|\vec{a}\times\vec{b}\|=\|\vec{a}\|\|\vec{b}\|\sin(\varphi). \] The direction of the vector $\vec{a}\times\vec{b}$ is perpendicular to both $\vec{a}$ and $\vec{b}$.