The example is :

Convert the scalar equation 4x − 2y + z = 2 into a parametric form.

This is not a big deal, usually. We have the coefficients, 4, -2 and 1, so the normal is

(4,−2,1)

(4,−2,1)=2 it’s not = 0, so there will be 2 vectors orthogonal

to it. Any two vectors that are not in the same direction will do. The best way to make sure they aren’t in the same direction is to make them cover different variables, i.e., use only x and y for one, y and z for the other.

For instance,

(1,2,0) is orthogonal to (4,−2,1) as is(0,1,2)

Those are clearly not in

the same direction. They’ll do. Recall that we need to match the right hand side. Any

combination of those two will not change the total on the left. We just need a single point that matches. So,

x =1/2-->(1/2,0,0)

y=-1--->(0,-1,0)

z=2--> ( 0,0,2)

So, any will do, here’s one final set

(xyz)=(1,1,0)+(1,2,0)t+(0,1,2)s

From all this my questions are when they mention that the normal is (4,-2,1) they say that the orthogonal is (1,2,0) and (0,1,2).

I was wondering how they got the orthogonal and how they got those two from the normal (4,-2,1)?

My second question is how did they get the final set?