Vector equation of a straight line in the space

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:$$(x,y,z)=(-1,1,3)+k\cdot (3,-2,1)$$