What is Perpendicular Distance from Origin to Plane? | Simple Explanation for Class 12

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What is the Perpendicular Distance from the Origin to a Plane?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

The perpendicular distance from the origin to a plane is the shortest possible distance from the point (0,0,0) to any point on that plane. Imagine drawing a straight line from the very center of your room (the origin) to a wall (the plane) such that the line hits the wall at a perfect 90-degree angle; that's the perpendicular distance.

Simple Example

Quick Example

Think of a large flat screen TV hanging on a wall in your living room. If the 'origin' is a specific corner of the room on the floor, the perpendicular distance from that corner to the TV screen would be the shortest straight line you could measure, hitting the screen directly at a right angle. It's not the distance to the top edge or a side corner, but the absolute closest point.

Worked Example

Step-by-Step

Let's find the perpendicular distance from the origin (0,0,0) to the plane given by the equation 3x + 4y - 5z = 10.

Step 1: Identify the coefficients of x, y, z and the constant term from the plane equation Ax + By + Cz = D.
Here, A = 3, B = 4, C = -5, and D = 10.

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Step 2: Use the formula for perpendicular distance from the origin to a plane: |D| / sqrt(A^2 + B^2 + C^2).

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Step 3: Substitute the values into the formula.
Distance = |10| / sqrt(3^2 + 4^2 + (-5)^2)

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Step 4: Calculate the squares and sum them.
Distance = 10 / sqrt(9 + 16 + 25)

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Step 5: Simplify the sum under the square root.
Distance = 10 / sqrt(50)

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Step 6: Simplify the square root. sqrt(50) = sqrt(25 * 2) = 5 * sqrt(2).
Distance = 10 / (5 * sqrt(2))

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Step 7: Further simplify the fraction.
Distance = 2 / sqrt(2)

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Step 8: Rationalize the denominator by multiplying numerator and denominator by sqrt(2).
Distance = (2 * sqrt(2)) / (sqrt(2) * sqrt(2)) = (2 * sqrt(2)) / 2 = sqrt(2).

Answer: The perpendicular distance from the origin to the plane is sqrt(2) units.

Why It Matters

This concept is super useful in fields like AI/ML, where it helps define how 'far' a data point is from a decision boundary. In engineering, it's used to calculate clearances for machinery or design optimal pathways for drones. Architects use it to ensure structural stability and precise measurements for buildings.

Common Mistakes

MISTAKE: Forgetting the absolute value around 'D' in the numerator. | CORRECTION: The distance must always be a positive value, so always take the absolute value of the constant term D.

MISTAKE: Incorrectly calculating the square root of the sum of squares, especially with negative numbers. | CORRECTION: Remember that (-C)^2 is always positive, like (-5)^2 = 25. Double-check your arithmetic for A^2 + B^2 + C^2.

MISTAKE: Mixing up the signs of A, B, C from the equation. | CORRECTION: Ensure you take the coefficients with their correct signs from the plane equation Ax + By + Cz + D = 0 or Ax + By + Cz = D. If the equation is 3x - 2y + z = 5, then A=3, B=-2, C=1.

Practice Questions

Try It Yourself

QUESTION: Find the perpendicular distance from the origin to the plane x + 2y - 2z = 6. | ANSWER: 2 units

QUESTION: What is the perpendicular distance from the origin to the plane 2x - 3y + 6z - 14 = 0? | ANSWER: 2 units

QUESTION: A plane passes through the point (1, -1, 2) and has normal vector (2, 1, -1). Find the equation of the plane and then calculate the perpendicular distance from the origin to this plane. | ANSWER: Plane equation: 2x + y - z + 1 = 0. Distance: 1 / sqrt(6) units.

MCQ

Quick Quiz

Which of the following formulas correctly calculates the perpendicular distance from the origin (0,0,0) to a plane Ax + By + Cz + D = 0?

D / sqrt(A^2 + B^2 + C^2)

|D| / sqrt(A^2 + B^2 + C^2)

sqrt(A^2 + B^2 + C^2) / |D|

(A+B+C) / D

The Correct Answer Is:

The formula for the perpendicular distance from the origin to a plane Ax + By + Cz + D = 0 is |D| / sqrt(A^2 + B^2 + C^2). The absolute value of D is crucial because distance must be positive.

Real World Connection

In the Real World

Imagine you're developing a navigation system for autonomous vehicles like self-driving cars or delivery drones in a city like Bengaluru. Knowing the perpendicular distance from the vehicle's current position (like the origin) to a virtual 'no-fly zone' plane or a 'safe path' plane helps the system maintain safe distances and avoid collisions, ensuring smooth and efficient travel.

Key Vocabulary

Key Terms

Origin: The point (0,0,0) in 3D space, often considered the starting point. | Plane: A flat, two-dimensional surface that extends infinitely in 3D space. | Perpendicular: At a right angle (90 degrees) to something. | Normal Vector: A vector that is perpendicular to a plane. | Coefficients: The numerical part of a term in an equation, like A, B, C in Ax + By + Cz = D.

What's Next

What to Learn Next

Great job understanding this! Next, you can learn about finding the perpendicular distance from *any* general point (not just the origin) to a plane. This builds directly on what you've learned and will unlock even more complex geometry problems!