What is the Test for Perpendicularity of a Line and a Plane?

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Grade Level:

Class 12

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Definition

What is it?

The test for perpendicularity of a line and a plane checks if a given line forms a 90-degree angle with every line in the plane that passes through their intersection point. Simply put, it tells us if a line stands straight up or down from a flat surface, like a flagpole from the ground.

Simple Example

Quick Example

Imagine a cricket stump standing perfectly upright on the pitch. If the stump represents a line and the cricket pitch represents a plane, the stump is perpendicular to the pitch. The test would confirm this perfect upright position.

Worked Example

Step-by-Step

Let's say we have a line L passing through point P(1, 2, 3) and having direction ratios (2, -1, 3). We also have a plane E given by the equation 4x - 2y + 6z = 10.

Step 1: Find the direction ratios of the line L. These are (a1, b1, c1) = (2, -1, 3).
---Step 2: Find the normal vector (direction ratios) of the plane E. From the equation Ax + By + Cz = D, the normal vector is (A, B, C). So, (a2, b2, c2) = (4, -2, 6).
---Step 3: For a line to be perpendicular to a plane, its direction ratios must be proportional to the normal vector's direction ratios of the plane. This means a1/a2 = b1/b2 = c1/c2.
---Step 4: Check the proportionality: 2/4 = -1/-2 = 3/6.
---Step 5: Simplify the ratios: 1/2 = 1/2 = 1/2.
---Step 6: Since the ratios are equal, the line L is perpendicular to the plane E.

Answer: The line is perpendicular to the plane.

Why It Matters

Understanding perpendicularity is key in designing stable structures, from buildings to robotic arms. Engineers use this to ensure things stand upright and are strong. It's vital in fields like AI/ML for 3D object recognition and in Physics for understanding force directions.

Common Mistakes

MISTAKE: Confusing perpendicularity of a line to a plane with perpendicularity of two lines. | CORRECTION: For line-plane perpendicularity, you compare the line's direction ratios with the PLANE'S NORMAL VECTOR'S direction ratios, not another line in the plane.

MISTAKE: Incorrectly extracting the normal vector from the plane's equation. | CORRECTION: For Ax + By + Cz = D, the normal vector is always (A, B, C). Make sure to include the correct signs for A, B, and C.

MISTAKE: Forgetting that the proportionality a1/a2 = b1/b2 = c1/c2 must hold true for ALL three ratios. | CORRECTION: Check all three ratios carefully. If even one pair is not equal, the line is not perpendicular to the plane.

Practice Questions

Try It Yourself

QUESTION: Is the line with direction ratios (3, 6, -9) perpendicular to the plane 2x + 4y - 6z = 15? | ANSWER: Yes

QUESTION: A line passes through the origin (0,0,0) and has direction ratios (1, 2, 3). Is this line perpendicular to the plane x + 2y - 3z = 7? | ANSWER: No

QUESTION: Find the value of 'k' if the line with direction ratios (k, 4, 2) is perpendicular to the plane 3x - 6y + 12z = 5. | ANSWER: k = -2

MCQ

Quick Quiz

Which condition must be true for a line with direction ratios (a1, b1, c1) to be perpendicular to a plane with normal vector (a2, b2, c2)?

a1a2 + b1b2 + c1c2 = 0

a1/a2 = b1/b2 = c1/c2

a1 = a2, b1 = b2, c1 = c2

a1b2 = a2b1

The Correct Answer Is:

For a line to be perpendicular to a plane, its direction vector must be parallel to the plane's normal vector. This means their direction ratios must be proportional, hence a1/a2 = b1/b2 = c1/c2. Option A is for perpendicular lines/vectors.

Real World Connection

In the Real World

When ISRO launches rockets, they need to ensure the rocket's trajectory is perpendicular to certain reference planes during launch to achieve stable flight. Also, architects designing multi-storey buildings in Mumbai use this concept to make sure pillars are perfectly upright and stable against the ground and each floor.

Key Vocabulary

Key Terms

PERPENDICULAR: Forming a 90-degree angle. | DIRECTION RATIOS: Numbers describing the direction of a line in 3D space. | NORMAL VECTOR: A vector that is perpendicular to a plane. | PROPORTIONAL: Having a constant ratio between corresponding parts.

What's Next

What to Learn Next

Great job understanding perpendicularity! Next, you should explore 'Angle Between a Line and a Plane'. This builds on what you've learned, helping you calculate angles when things aren't perfectly straight, which is super useful in many real-world designs.