What is the Test for Perpendicularity of Two Planes?

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Grade Level:

Class 12

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Definition

What is it?

The test for perpendicularity of two planes checks if they meet at a right angle (90 degrees). We use their normal vectors to determine this. If the dot product of their normal vectors is zero, the planes are perpendicular.

Simple Example

Quick Example

Imagine two walls in your room meeting at a perfect corner. Those walls are perpendicular. In maths, we check if the 'direction arrows' (normal vectors) coming straight out from each wall are at 90 degrees to each other. If they are, the walls are perpendicular.

Worked Example

Step-by-Step

Let Plane 1 be given by the equation: 2x + 3y - z = 5. --- Let Plane 2 be given by the equation: 4x - 2y + 2z = 7. --- Step 1: Find the normal vector for Plane 1. The coefficients of x, y, and z give the normal vector. So, n1 = (2, 3, -1). --- Step 2: Find the normal vector for Plane 2. So, n2 = (4, -2, 2). --- Step 3: Calculate the dot product of n1 and n2. n1 . n2 = (2)(4) + (3)(-2) + (-1)(2). --- Step 4: Perform the multiplication and addition. n1 . n2 = 8 - 6 - 2. --- Step 5: Calculate the final value. n1 . n2 = 0. --- Step 6: Since the dot product is 0, the two planes are perpendicular. | ANSWER: The planes are perpendicular.

Why It Matters

Understanding perpendicular planes is crucial in engineering for designing stable structures like bridges and buildings. In computer graphics, it helps create realistic 3D models and animations for games and movies. Architects and civil engineers use this concept daily.

Common Mistakes

MISTAKE: Confusing the plane's equation with its normal vector. Students sometimes use the constant term or variables directly. | CORRECTION: The normal vector is formed ONLY by the coefficients of x, y, and z in the plane's general equation (Ax + By + Cz = D), so n = (A, B, C).

MISTAKE: Incorrectly calculating the dot product, especially with negative numbers. | CORRECTION: Remember the formula for dot product: (a1, b1, c1) . (a2, b2, c2) = a1*a2 + b1*b2 + c1*c2. Pay close attention to signs during multiplication and addition.

MISTAKE: Assuming planes are perpendicular if their equations look similar or have some matching coefficients. | CORRECTION: Always perform the dot product test. Visual inspection can be misleading; the mathematical test is definitive.

Practice Questions

Try It Yourself

QUESTION: Are the planes x + y + z = 10 and 2x - y - z = 5 perpendicular? | ANSWER: No

QUESTION: Determine if the plane 3x - 4y + 5z = 8 is perpendicular to the plane -6x + 8y - 10z = 12. | ANSWER: Yes (because their normal vectors are parallel, meaning they are the same plane or parallel planes, which are considered perpendicular to a third plane if that third plane is perpendicular to either of them. Here, the question implies checking perpendicularity of the *given* two planes, which are actually parallel, not perpendicular. Let's re-evaluate for a true perpendicular case. For 3x - 4y + 5z = 8 and 4x + 3y = 7. n1=(3,-4,5), n2=(4,3,0). Dot product = 12 - 12 + 0 = 0. So, for 3x - 4y + 5z = 8 and 4x + 3y = 7, the answer is YES.)

QUESTION: Find the value of 'k' such that the plane 2x - ky + 3z = 1 and the plane 4x + 2y - 2z = 5 are perpendicular. | ANSWER: k = 1

MCQ

Quick Quiz

If the normal vectors of two planes are n1 and n2, what condition must be met for the planes to be perpendicular?

n1 + n2 = 0

n1 . n2 = 0

n1 x n2 = 0

n1 = n2

The Correct Answer Is:

For two planes to be perpendicular, their normal vectors must be perpendicular. The dot product of two perpendicular vectors is always zero. Options A, C, and D do not represent this condition.

Real World Connection

In the Real World

When ISRO launches satellites, the different stages of the rocket separate precisely. The design of these separation mechanisms, and ensuring the rocket's trajectory is stable, involves understanding how forces and surfaces interact at right angles. Also, when you build a house, the floor and walls, or two walls meeting, must be perpendicular for a strong and stable structure. Civil engineers use this concept to ensure buildings stand upright and are safe.

Key Vocabulary

Key Terms

PLANE: A flat, two-dimensional surface that extends infinitely in all directions. | NORMAL VECTOR: A vector perpendicular to a plane, indicating its orientation. | DOT PRODUCT: A mathematical operation on two vectors that results in a scalar (a single number). | PERPENDICULAR: Meeting or crossing at a right angle (90 degrees).

What's Next

What to Learn Next

Great job learning about perpendicular planes! Next, you can explore the 'Test for Parallelism of Two Planes'. It also uses normal vectors but checks for a different relationship between planes. Understanding both will give you a complete picture of how planes interact in 3D space.