Is there a rule regarding how vectors and scalars combine in formulae? $$\begin{align}
\frac{\vec{d}}{t} &= \vec{v} & \frac{\text{vector}}{\text{scalar}} &= \text{vector} \\
\vec{d}\times \vec{d} &= \vec{A} & \text{vector}\times\text{vector} &= \text{vector} \\
\vec{d}\times \vec{d}\times \vec{d} &= V & \text{vector}\times\text{vector}\times\text{vector} &= \text{scalar} \\
\frac{\vec{F}}{\vec{A}} &= P & \frac{\text{vector}}{\text{vector}} &= \text{scalar}
\end{align}$$
Above are some examples of how various mathematical combinations of vectors and scalars produce a vector or a scalar.
Is there some general rule independent of the quantities involved or must each situation be treated individually ?
 A: If in general by division we mean the 'inverse operations' of the product, the only possible 'division between vectors' can be defined starting from the vector product definition. Given two vectors $\vec a$ and $\vec b$ we define the vector product as:
$$\vec a \times \vec b= \hat n \ ab \sin θ \tag 1$$
where $\hat n$ is a normal unit vector both $\vec a$ and $\vec b$ and $θ$ is the angle formed by two vectors. The definition of division between vectors as 'inverse operation' of the vector product necessarily implies two 'limitations'...
(a) the two vectors must be normal to each other;
(b) the resulting 'quotient vector' is a function of the choice of $θ$ in $(1)$, in the sense that it is different for different values of $θ$ and is therefore not uniquely determined.
A: Like we never find $\mathrm{d}x$ in a denominator of an integral
$$\int \frac{f(x)}{\mathrm{d}x} $$
we  never find a vector-valued quantity in denominator
$$\frac{\text{whatever}}{\vec{A}}$$
But don't confuse it with the following legal operations
$$\frac{\text{whatever}}{\vec{A}\cdot \vec{B}}$$
or
$$\frac{\text{whatever}}{|\vec{A}|}$$
where the denominators are no longer vector valued quantities.
A: It is very important to know and be careful about which operations you are using.

*

*Vectors can be multiplied by scalars so this operation is valid.


*The cross product of two vectors produces another vector so this is ok.


*The cross product is not associative so you have to be careful about which order you're performing these cross products in. The cross product of two vectors produces another vector so an operation such as $\vec v\times \vec v\times \vec v$ as you've written is valid but the result is a vector, not a scalar.


*Division of one vector by another is not a defined operation.
Given a 3-dimensional vector space $V$ defined over a scalar field $\Bbb F$ (either $\Bbb R$ or $\Bbb C$) the following important operations are defined (not an exhaustive list):
Vector Addition: $$+:V \times V\rightarrow V$$
Scalar Multiplication of Vectors: $$\bullet:V\times \Bbb F\rightarrow V$$
The Dot Product: $$\cdot:V\times V\rightarrow \Bbb F$$
The Cross Product: $$\times: V\times V\rightarrow V$$
A: There is an extension of vectors and scalars called a 'Geometric Algebra' or 'Clifford Algebra' that allows vectors and scalars to be added and multiplied together freely, where multiplication is generally invertible (so division of vectors can be defined), and which includes the usual dot and cross products of vector algebra as special cases. So you can use Geometric Algebra to provide general rules for combining vectors and scalars, and use them to derive the rules for dot and cross products.
A Geometric Algebra for 3D space contains 8 dimensions, one scalar, three for vectors (directed lengths), three for bivectors (a new sort of object representing oriented surface areas), and one for a trivector (representing an oriented volume). You can construct objects, called multivectors, by adding any linear combination of these. So a scalar can be added to a vector, to produce a 'paravector'. A scalar can be added to a bivector to produce a quaternion (or in 2D, a complex number).
The Geometric Algebra multiplies multivectors using the geometric product. If you multiply two vectors together, you get a linear combination of a scalar (equal to the dot product of the vectors) and a bivector (which is dual to the cross product of the vectors). Similarly, you can define division of vectors, which gives a combination of a scalar and a bivector. If the vectors are parallel, only the scalar part is non-zero. If the vectors are perpendicular, only the bivector part is non-zero. Otherwise you get a combination of scalar and bivector.
Geometric Algebra can be defined in spaces of any dimension. In the Geometric Algebra for 4D space, we have one dimension for scalars, four for vectors, six for bivectors, four for trivectors, and one for quadvectors. This is the reason why cross products only work in 3D space. With three dimensions, vectors and bivectors have the same dimension, and you can flip the bivector into a (mostly) equivalent vector, using the dual operation. This is sometimes called an 'axial vector', because it has different transformation properties to the normal 'polar vector'. In 4D, bivectors have six dimensions, and so you can't make a 4D vector out of it by taking the dual. This example shows how using a more general unifying rule can be used to derive and explain features of how the rules for dot and cross products work, and why they are the way they are.
There are plenty of basic introductions to Geometric Algebra on the web.
