Plane normal and distance from a point

Parametric equations

The plane with normal vector $\mathbf{n}=(a,b,c)\ne 0$ through $P_0(x_0,y_0,z_0)$:

Expanded, this is the general form $ax+by+cz=h$, where $h=a{x}_0+b{y}_0+c{z}_0$. The distance from a point $P_1(x_1,y_1,z_1)$ to the plane is

Multiplying $(a,b,c,h)$ by a nonzero constant does not change the plane. It only changes the scale of the equation.

Interaction

• Plane tilt: rotate the unit normal $\hat{\mathbf{n}}$. The plane turns with it, and the coefficients $(a,b,c)$ update together
• Plane offset h: translate the plane. The foot slides along the normal, and the distance changes
• Test point height: move $P_1$ and compare the sign of the signed distance, which marks which side of the normal it is on
• Equation scale k: multiply by a nonzero constant. The coefficients all change, while the plane, the foot, and the distance stay put
• Drag the figure to rotate the view and check that the perpendicular segment is orthogonal to the plane. Horizontal view and Elevation do the same rotation

What to notice

• The normal vector is orthogonal to every displacement in the plane. Multiplying $(a,b,c,h)$ by a nonzero constant does not change the plane. It only changes the scale of the equation.
• $h=a{x}_0+b{y}_0+c{z}_0$ encodes which point the plane passes through. Changing $P_0$ translates the plane.
• The distance from a point to a plane is a direct use of the dot product. It extends the projection and normal reading of plane vectors into space.

Related

• Spatial vector and three plane projections
• Where a line meets a plane
• Geometric significance of the dot product
• Spatial vectors, planes, and lines

Further reading

• Euclidean plane (Wikipedia)
• Normal (geometry) (Wikipedia)
• Distance from a point to a plane (Wikipedia)