What is a direction vector?

Hint: Any mathematical representation of physical quantities that requires a definition of both, magnitude and direction is called a vector. A vector quantity has a magnitude and a certain direction. We can understand this by the example that, “the car is moving at a speed of twenty meters per second towards the North direction” is a vector statement.

Complete step-by-step answer:
A direction vector is a unit vector in the direction of a particular vector. The term unit vector is used because the magnitude of this direction vector is unity.
In the Cartesian system of coordinates, there are three-unit vectors. These are $\widehat{i}$ along the X-axis, $\widehat{j}$ along the Y-axis and $\widehat{k}$ along the Z-axis. We can use these three-unit vectors to write the equation of any vector in the Cartesian coordinate system. The generalized form of writing a vector in Cartesian system is:
$\Rightarrow \overrightarrow{R}=a\widehat{i}+b\widehat{j}+c\widehat{k}$
Where,
‘R’ is the vector to be represented.
‘a’ is the magnitude of a vector along the X-axis.
‘b’ is the magnitude of a vector along the Y-axis.
‘c’ is the magnitude of a vector along the Z-axis.

And the generalized form of writing a direction vector along the direction of above vector is:
$\Rightarrow \widehat{r}=\dfrac{a\widehat{i}+b\widehat{j}+c\widehat{k}}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}$
Hence, the direction is defined and its generalized formula has been derived.

Note: Any vector in space is unchanged when shifted parallel to itself to some other point in space. This means on parallel shifting the vector from one point to another, its magnitude and direction remains the same. Also, vector operation of two or more vectors can be done using basic addition and subtraction. Apart from these operators, vectors can be operated by their dot product and cross product.