Finding the scalar equation of a line

Formulas for the scalar equation of a line

To find the scalar equation of a line, we’ll use the formulas

$x=x_0+at$

$y=y_0+bt$

$z=z_0+ct$

where $P_0(x_0,y_0,z_0)$ is a given point and $v=\langle a,b,c\rangle$ is the given vector. The vector may also be in the format $v=ai+bj+ck$ .

Example

Find the scalar equation of the line.

$P(2,3,1)$

$\langle2,-1,5\rangle$

Plugging the given point and the given vector into our formulas for $x$ , $y$ and $z$ , we get

$x=2+2t$

$y=3-t$

$z=1+5t$

The scalar equation of the line is given by $x=2+2t$ , $y=3-t$ and $z=1+5t$ .