Proving a line lies entirely on a plane

[barren birch]

Proving the point les on the plane is fine, but the second part of this question is throwing me off a bit. Do I need to prove that the line is also parallel to the plane? If so, I have FP1 vectors knowledge so if you can, can you use the cross product somewhere or maybe the dot product to prove the second part? <@&791435371564892232>

[rigid notch]

Find a (mu, lambda) for every t is the most intuitive way

[main thicket]

use cross product rule to find the equation of the cartesian plane and then use the equation of the line to sub it into the plane

if you get for instance 15=15 23=23 than it will lie entirely in the plane

[hexed edge]

as an alternative to the above method

1. find the equation of the plane in the form r.n=k using cross product
2. show that n is perpendicular to the direction vector of the line using dot product.

[rigid notch]

This isn’t sufficient, you would have to show that the position vector of the line also lies on the plane

[rigid notch]

If you don’t want to cross product, you can sub in lambda = 3t-2 and mu = t+1 which comes out as the same equation as the line.