Distance of Origin from a Plane
Three-Dimensional Geometry · Class XII
Plane ax + by + cz = d. Distance from origin is |d| / √(a² + b² + c²).
- Normal magnitude3
- Distance from O2
- Foot-of-perp x0.6667
- Distance from O as d slides
Formulas in this lab
- Normal magnitude
- Distance from O
- Foot-of-perp x
Frequently asked questions
▶What is the distance from a point to a plane?
For the plane $ax + by + cz + d = 0$ and the point $(x_0, y_0, z_0)$, the perpendicular distance is $\dfrac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2+b^2+c^2}}$. The absolute value ensures the distance is non-negative.
▶How do I use the point-to-plane distance lab?
Slide the plane coefficients $a, b, c, d$ and the point coordinates. The lab returns the distance live. Try the plane $x + y + z = 3$ and the origin: distance $= 3/\sqrt{3} = \sqrt{3} \approx 1.73$.
▶Why is the absolute value crucial?
Without it, the sign tells you which side of the plane the point lies on — useful for half-space arguments but wrong for a distance, which must be non-negative. JEE often asks for the side as well, separately from the distance. The lab outputs the unsigned distance.
▶Where does this formula appear in real life?
Computer graphics uses it for collision detection: is a player on the right side of a wall? Linear regression's residuals are signed distances from data points to the best-fit plane (hyperplane in higher dimensions). Support Vector Machines maximize the margin computed by this formula.