To determine a straight line in space we need a point and a direction. Any vector that has the same direction as a given straight line is a director vector of the above mentioned straight line.
It is worth mentioning that like on the plane for any given two points we can have a point and a vector and vice versa.
Let's consider in the reference system $$\{O; \overrightarrow{i},\overrightarrow{j},\overrightarrow{k} \}$$ the straight line $$r$$ that goes through point $$A$$ and has a director vector $$\overrightarrow{v}$$. We will symbolize it by $$r\Big(A;\overrightarrow{v}\Big)$$.
There are different ways of expressing it. Let's see now the vectorial form.
Given a point $$P$$ of a straight line, we can express it by:$$$P=A+k\cdot \overrightarrow{v}$$$This expression is known as the vector equation of the straight line.
If we want to specify the coordinates in the space:$$$(x,y,x)=(a_1,a_2,a_3)+k\cdot (v_1,v_2,v_3)$$$
Given the point $$A = (-1, 1, 3)$$ and the vector $$\overrightarrow{v}=(3,-2,1)$$, find the vector equation that starts at point $$A$$ and has the direction of the vector $$\overrightarrow{v}$$.
From the formula$$$P=A+k\cdot \overrightarrow{v}$$$we get:$$$(x,y,z)=(-1,1,3)+k\cdot (3,-2,1)$$$