How to bring vec equation for plane to the correct form?

Question

How to bring vec equation for plane to the correct form?

I have:

$r \cdot (4i-j-2k)=4*0+(-1)*2+(-2)*(-3)$

$r \cdot (4i-j-2k)=4*0+(-1)*2+(-2)*(-3)$

$r \cdot (4i-j-2k)=4$

What to do next?

Perhaps,

$(x,y,z)\cdot(4,-1,-2)=4$

$4x-y-2z=4$

Answer 1

Your result is correct but the notation is bad. It is better to write all the vectors in the same ''components'' notation as:

$$ (x,y,z)^T\cdot(4,-1,-2)^T=4 \quad \iff \quad 4x-y-2z=4 $$