Write parametric equations for the line through the distinct points P(x1, y1, z1) and Q(x2, y2, z2).
The parametric equations for a line in 3D space can be written as, (using P)
`x = x_1+at`
`y = y_1+bt`
`z = z_1+ct`
Now Q is also on the same line (with a different parameter value)
`x_2 = x_1+at'`
`y_2 = y_1+bt'`
`z_2 = z_1+ct'`
Solving for a using both x equations,
`(x-x_1)/t = (x_2-x_1)/(t')`
Now this gives,
`(x-x_1)/(x_2-x_1) = t/(t') = m`
From y and z equations also,
`(y-y_1)/(y_2-y_1) = t/(t') = m`
`(z-z_1)/(z_2-z_1) = t/(t') = m`
Therefore we write the two point formula as below,
`(x-x_1)/(x_2-x_1) = (y-y_1)/(y_2-y_1) = (y-y_1)/(y_2-y_1) = m`
Rearranging the above three equations will give us the parametric equation with `m` as a parameter.
`x = x_1+m(x_2-x_1)`
`y = y_1+m(y_2-y_1)`
`z = z_1+m(z_2-z_1)`
These are the required equations.