Distance between a straight line and a plane in space

Notice the relative positions between a straight line $r$ and a plane $π$ to calculate the distance between them:

If the straight line is included in the plane or if the straight line and the planes are secant, the distance between both is zero, $d (r, π) = 0$
If the straight line and the plane are parallel, the distance between both is calculated taking a point $P$ of the straight line and calculating the distance between $P$ and the plane. $d (r, π) = d (P, π) where P \in r$

Example

Find the distance between the straight line $r : x - 2 = y = z + 1$ and the plane $π : x + y - 2 z + 3 = 0$ .

We verify that the plane and the straight line are parallel using the scalar product between the governing vector of the straight line, $\vec{v}$ , and the normal vector of the plane $\vec{n}$ . If the straight line and the plane are parallel the scalar product will be zero: $\vec{v} \cdot \vec{n} = (1, 1, 1) \cdot (1, 1, - 2) = 1 + 1 - 2 = 0$

So they are parallel. We look for a point of the straight line, $Q = (2, 0, - 1)$ , and apply the formula: $d (r, π) = d (P, π) = \frac{| 1 \cdot 2 + 1 \cdot 0 - 2 \cdot (- 1) + 3 |}{\sqrt{1^{2} + 1^{2} + (- 2)^{2}}} = \frac{7}{\sqrt{6}}$