Find the vector and the cartesian equations of the lines that passes through the origin and (5, –2, 3).

We know that

The vector equation of as line which passes through two points whose position vectors are and is .

Here, the position vectors of the two points (0, 0, 0) and (5, -2, 3) are and , respectively.

So, The vector equation of the required line is:

Now, we know that

Cartesian equation of a line that passes through two points (x1, y1, z1) and (x2, y2, z2) is

So, the Cartesian equation of the line that passes through the origin (0, 0, 0) and (5, – 2, 3) is