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 ${O; \vec{i}, \vec{j}, \vec{k}}$ the straight line $r$ that goes through point $A$ and has a director vector $\vec{v}$ . We will symbolize it by $r (A; \vec{v})$ .

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: $P = A + k \cdot \vec{v}$ This expression is known as the vector equation of the straight line.

If we want to specify the coordinates in the space: $(x, y, x) = (a_{1}, a_{2}, a_{3}) + k \cdot (v_{1}, v_{2}, v_{3})$

Example

Given the point $A = (- 1, 1, 3)$ and the vector $\vec{v} = (3, - 2, 1)$ , find the vector equation that starts at point $A$ and has the direction of the vector $\vec{v}$ .

From the formula $P = A + k \cdot \vec{v}$ we get: $(x, y, z) = (- 1, 1, 3) + k \cdot (3, - 2, 1)$