Chapter 1, Problem 20P

The parametric equations of a line in space are: $x=x_0+at,y=y_0+bt,\text{and}z\text{ = }{z}_{\text{0}}+ct.$ The distance d from a point $A\left(x_A,y_A,z_A\right)$ to the line can be calculated by:

$d=d_{A0}\sin \left[\text{a}\cos \left(\frac{\left(x_A-x_0\right)a+\left(y_A-y_0\right)b+\left(z_A-z_0\right)c}{d_{A0}\sqrt{a^2+b^2+c^2}}\right)\right]$

where $d_{A0}=\sqrt{{\left(x_A-x_0\right)}^2+{\left(y_A-y_0\right)}^2+{\left(z_A-z_0\right)}^2},$

Determine the distance of the point A (2, -3, 1) from the line $x=-4+0.6t,y=-2+0.5t,$ and $z=-3+0.7t$ . First define the variables $x_0,y_0,z_0,a,b,\text{and}c$ , then use the variable (and the coordinates of point A) to calculate the variable d_A0, and finally calculate d.