What is the Distance Between a Point and a Plane?

S7-SA2-0286

Grade Level:

Class 12

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

Definition

What is it?

The distance between a point and a plane is the shortest possible length of a line segment that connects the given point to any point on the plane. Imagine dropping a straight plumb line from the point directly down to the plane; the length of that line is the distance we are looking for.

Simple Example

Quick Example

Imagine your mobile phone (the point) is floating in the air, and the floor of your room is the plane. The shortest distance from your phone to the floor is a straight line going directly down. If your phone is 2 meters above the floor, then 2 meters is the distance between the point (your phone) and the plane (the floor).

Worked Example

Step-by-Step

Let's find the distance between the point P(1, 2, 3) and the plane given by the equation 2x + 3y + 4z - 5 = 0.

1. Identify the coordinates of the point (x1, y1, z1) and the coefficients of the plane equation (A, B, C, D).
Point P(1, 2, 3) means x1=1, y1=2, z1=3.
Plane equation 2x + 3y + 4z - 5 = 0 means A=2, B=3, C=4, D=-5.
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2. Use the formula for the distance between a point (x1, y1, z1) and a plane Ax + By + Cz + D = 0:
Distance = |Ax1 + By1 + Cz1 + D| / sqrt(A^2 + B^2 + C^2)
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3. Substitute the values into the formula:
Distance = |(2)(1) + (3)(2) + (4)(3) + (-5)| / sqrt(2^2 + 3^2 + 4^2)
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4. Calculate the numerator:
Numerator = |2 + 6 + 12 - 5| = |15|
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5. Calculate the denominator:
Denominator = sqrt(4 + 9 + 16) = sqrt(29)
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6. Divide the numerator by the denominator:
Distance = 15 / sqrt(29)
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7. The distance between the point P(1, 2, 3) and the plane 2x + 3y + 4z - 5 = 0 is 15 / sqrt(29) units.

Why It Matters

Understanding this distance is crucial in fields like robotics and AI, where robots need to navigate without hitting obstacles (planes). In engineering, it helps design safe structures and plan routes for drones or self-driving cars. Even in medical imaging, doctors use similar concepts to measure distances inside the human body.

Common Mistakes

MISTAKE: Forgetting the absolute value in the numerator. | CORRECTION: The distance is always a positive value, so always take the absolute value of the expression (Ax1 + By1 + Cz1 + D).

MISTAKE: Incorrectly identifying the 'D' term in the plane equation. | CORRECTION: Ensure the plane equation is in the form Ax + By + Cz + D = 0. If it's Ax + By + Cz = D', then D will be -D'. For example, if it's 2x+3y+4z=5, then D = -5.

MISTAKE: Making calculation errors in the square root of the denominator. | CORRECTION: Carefully square each coefficient (A, B, C) and then add them before taking the square root. Double-check your arithmetic.

Practice Questions

Try It Yourself

QUESTION: Find the distance between the point (0, 0, 0) and the plane 3x + 4y + 5z - 10 = 0. | ANSWER: 10 / sqrt(50) or 10 / (5 * sqrt(2)) = 2 / sqrt(2) = sqrt(2) units.

QUESTION: A point is at (2, -1, 4) and a plane is given by x - 2y + 2z + 7 = 0. What is the distance? | ANSWER: 19 / 3 units.

QUESTION: If the distance between the point (1, 1, k) and the plane 2x + 3y - 4z + 1 = 0 is 1 unit, find the possible values of k. | ANSWER: k = 1 + sqrt(29)/4 or k = 1 - sqrt(29)/4

MCQ

Quick Quiz

Which of the following is the correct formula for the distance between a point (x1, y1, z1) and a plane Ax + By + Cz + D = 0?

(Ax1 + By1 + Cz1 + D) / (A^2 + B^2 + C^2)

|Ax1 + By1 + Cz1 + D| / sqrt(A^2 + B^2 + C^2)

sqrt((Ax1)^2 + (By1)^2 + (Cz1)^2 + D^2)

(Ax1 + By1 + Cz1 + D) / sqrt(A + B + C)

The Correct Answer Is:

Option B correctly uses the absolute value in the numerator to ensure a positive distance and the square root of the sum of squares of coefficients in the denominator, which is the magnitude of the normal vector to the plane. Other options are mathematically incorrect for distance calculation.

Real World Connection

In the Real World

Imagine you are using a mapping app like Google Maps or Ola Cabs on your phone. When a drone delivers a package, its navigation system constantly calculates the distance between the drone (a point) and various surfaces like buildings, trees, or the ground (planes) to avoid collisions and find the safest path. This helps ensure your package arrives safely and quickly!

Key Vocabulary

Key Terms

PLANE: A flat, two-dimensional surface that extends infinitely in all directions. | NORMAL VECTOR: A vector perpendicular to a plane, whose components are the coefficients A, B, C in the plane equation. | ABSOLUTE VALUE: The non-negative value of a number, ignoring its sign. | COORDINATES: A set of numbers that defines the position of a point in space.

What's Next

What to Learn Next

Great job understanding this! Next, you can explore the 'Distance Between Two Parallel Planes'. This builds on what you've learned by applying similar ideas to find the distance between two flat surfaces that never meet, which is super useful in architecture and design!