Using Cross-Product and Vectors to find the distance between parallel lines ?

[Beamsbox]

Given the following two lines, prove that they are parallel, then find the distance between them...

(I have circled in red the two parts of the answer which I don't understand. Namely, why are they using the cross-product here, doesn't that give you a value perpendicular to the lines, hence a new vector that is on the parallel planes of the lines? The second part I highlighted I have no idea why they chose this.)

Could someone please explain the reasons to me? Something that I could visualize would be helpful.

Thanks, prior!

 

[LCKurtz]

Remember the formula

$$\| \vec{AP}\times\vec{AB}\| =\|\vec{AP}\|\|\vec{AB}\|\sin(\theta)$$

so your formula results in
$$\|\vec{AP}\|\sin(\theta)$$

Draw a picture of two parallel lines and label A,P, and B and the angle θ between the vectors AP and AB. If you draw a perpendicular line between the vectors you will see that is its length by looking at the triangle with θ the angle between the two vectors.