Plane and Line Distance

Question

How it comes that |D| is the distance of a plane to the origin?

Answer 1

$n$ is the normal vector whose direction is perpendicular to the place, i.e. the same direction in which we measure distance. The magnitude of $n$ is not really important here. All we need is the magnitude of the projection of $p$ on $n$ which is given by the absolute value of the scalar product of $p$ and $n$, which is $$p^{\prime}n=(4,0,0)^{\prime}(1,1,1)=4 \times 1 + 0 \times 1 + 0 \times 1=4$$ Hence the required distance. The picture presented there is helpful in understanding why the projection leads to the distance. Hope it helps.

Answer 2

As distance can never be negative as in your case -4. So they take magnitude of D. So D changes to |D|.

Answer 3

For the plane $$ax+by+cz=D$$ the distance from the origin to the plane is $$d=\left|\frac{D}{\sqrt{a^2+b^2+c^2}}\right|$$