What is the Test for Perpendicularity of Two Lines?

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Grade Level:

Class 12

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Definition

What is it?

The test for perpendicularity of two lines helps us check if two lines meet each other at a perfect 90-degree angle. This is like checking if the corner of a room or a cricket pitch boundary is perfectly square. We use their direction ratios or slopes to find this out.

Simple Example

Quick Example

Imagine your mobile phone screen. The top edge and the side edge meet at a perfect right angle. If you draw two lines representing these edges, they are perpendicular. The test for perpendicularity helps us mathematically confirm if any two lines, like the edges of a photo frame, form such a 90-degree angle.

Worked Example

Step-by-Step

Let's check if line L1 passing through A(1, 2, 3) and B(3, 5, 1) is perpendicular to line L2 passing through C(0, 1, 0) and D(2, 0, 3).

Step 1: Find the direction ratios for Line L1. For L1, (x1, y1, z1) = (1, 2, 3) and (x2, y2, z2) = (3, 5, 1).
Direction ratios (a1, b1, c1) = (x2-x1, y2-y1, z2-z1) = (3-1, 5-2, 1-3) = (2, 3, -2).
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Step 2: Find the direction ratios for Line L2. For L2, (x1, y1, z1) = (0, 1, 0) and (x2, y2, z2) = (2, 0, 3).
Direction ratios (a2, b2, c2) = (x2-x1, y2-y1, z2-z1) = (2-0, 0-1, 3-0) = (2, -1, 3).
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Step 3: Apply the perpendicularity condition: a1*a2 + b1*b2 + c1*c2 = 0.
Substitute the direction ratios: (2)*(2) + (3)*(-1) + (-2)*(3).
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Step 4: Calculate the sum.
4 - 3 - 6 = 1 - 6 = -5.
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Step 5: Check if the sum is 0.
Since -5 is not equal to 0, the lines L1 and L2 are NOT perpendicular.

Answer: The lines are not perpendicular.

Why It Matters

Understanding perpendicularity is key in many advanced fields. In Engineering, it helps design strong buildings and perfect roads. In Computer Graphics, it ensures objects are placed correctly in 3D games. Even in Robotics, it's used to make robot arms move precisely, avoiding collisions.

Common Mistakes

MISTAKE: Confusing perpendicularity with parallelism. Students sometimes use the condition for parallel lines (direction ratios are proportional) instead of perpendicularity. | CORRECTION: Remember, for perpendicular lines, the dot product of their direction ratios must be zero (a1a2 + b1b2 + c1c2 = 0). For parallel lines, direction ratios are proportional (a1/a2 = b1/b2 = c1/c2).

MISTAKE: Forgetting to find direction ratios first if only points are given. Students might directly use point coordinates in the perpendicularity formula. | CORRECTION: Always calculate the direction ratios of each line using the given points (x2-x1, y2-y1, z2-z1) before applying the perpendicularity test.

MISTAKE: Making calculation errors with negative numbers. A simple sign mistake can lead to a wrong conclusion. | CORRECTION: Double-check all multiplications and additions, especially when dealing with negative direction ratios, to ensure the final sum is accurate.

Practice Questions

Try It Yourself

QUESTION: Are the lines with direction ratios (1, 2, 3) and (-2, 1, 0) perpendicular? | ANSWER: No

QUESTION: Line L1 passes through (0,0,0) and (1,1,1). Line L2 passes through (1,0,0) and (0,1,-1). Are L1 and L2 perpendicular? | ANSWER: Yes

QUESTION: If a line has direction ratios (k, 2, -1) and is perpendicular to a line with direction ratios (3, -1, 4), find the value of k. | ANSWER: k = 2

MCQ

Quick Quiz

What is the condition for two lines with direction ratios (a1, b1, c1) and (a2, b2, c2) to be perpendicular?

a1/a2 = b1/b2 = c1/c2

a1 + a2 = b1 + b2 = c1 + c2

a1*a2 + b1*b2 + c1*c2 = 0

a1*a2 * b1*b2 * c1*c2 = 1

The Correct Answer Is:

Option C correctly states that for two lines to be perpendicular, the sum of the products of their corresponding direction ratios must be zero. This is derived from the dot product of their direction vectors. Options A, B, and D are incorrect conditions.

Real World Connection

In the Real World

Imagine an ISRO engineer designing a satellite. They need to ensure different solar panels or antennas are perfectly perpendicular to each other for optimal signal reception or power generation. Using the test for perpendicularity, they can mathematically confirm these alignments, preventing costly errors once the satellite is in space.

Key Vocabulary

Key Terms

PERPENDICULAR: Meeting at a right angle (90 degrees) | DIRECTION RATIOS: Numbers proportional to the direction cosines of a line, indicating its direction in space | DOT PRODUCT: A mathematical operation on two vectors that results in a single scalar value | RIGHT ANGLE: An angle of 90 degrees, like the corner of a square

What's Next

What to Learn Next

Great job learning about perpendicular lines! Next, you can explore the 'Test for Parallelism of Two Lines' to understand how to check if lines run side-by-side without ever meeting. This will help you fully understand how lines behave in 3D space!