The Condition for Perpendicularity

Oct 13, 2025 by miodeya.com 3 views

Introduction

In geometry, two lines are said to be perpendicular if they intersect at a right angle. In this article, we will explore how to find the value of p such that the given lines are perpendicular to each other. We will use the concept of the slope of a line and the condition for perpendicularity to solve this problem.

The Condition for Perpendicularity

Two lines are perpendicular if and only if the product of their slopes is -1. The slope of a line in the form y = mx + c is m. Therefore, if we have two lines with slopes m1 and m2, they are perpendicular if and only if m1 * m2 = -1.

Finding the Slopes of the Given Lines

To find the value of p such that the given lines are perpendicular, we need to find the slopes of the lines. The given lines are:

$\frac{4-x}{6}=\frac{7y+14}{8p}=\frac{z-5}{-4}$

and

$\frac{7+7x}{p}=\frac{y-1}{2}=\frac{z-4}{-2}$

We can rewrite the first line as:

$\frac{4-x}{6}=\frac{7y+14}{8p}$

$\frac{4-x}{6}=\frac{7(y+2)}{8p}$

$\frac{4-x}{6}=\frac{7y+14}{8p}$

Cross-multiplying, we get:

$8p(4-x) = 6(7y+14)$

$32p-8px = 42y+84$

$-8px = 42y+84-32p$

$-8px+32p = 42y+84$

$-8x+32 = 42y+32p$

Dividing both sides by -8, we get:

$x-\frac{32}{8} = \frac{42y+32p}{-8}$

$x-4 = \frac{42y+32p}{-8}$

$\frac{x-4}{-1} = \frac{42y+32p}{-8}$

$\frac{x-4}{-1} = \frac{21y+16p}{-4}$

Now, we can rewrite the second line as:

$\frac{7+7x}{p}=\frac{y-1}{2}$

Cross-multiplying, we get:

$2(7+7x) = p(y-1)$

$14+14x = py-p$

$14+14x = p(y-1)$

$14+14x = py-p$

$14+14x+1 = py$

$15+14x = py$

Dividing both sides by p, we get:

$\frac{15+14x}{p} = y$

Finding the Slopes of the Lines

Now that we have rewritten the lines, we can find the slopes of the lines. The slope of the first line is:

m1 = -\frac{21}{4}

The slope of the second line is:

m2 = \frac{1}{p}

The Condition for Perpendicularity

Two lines are perpendicular if and only if the product of their slopes is -1. Therefore, we can set up the following equation:

m1 * m2 = -1

$-\frac{21}{4} * \frac{1}{p} = -1$

$-\frac{21}{4p} = -1$

Multiplying both sides by -4p, we get:

21 = 4p

Dividing both sides by 4, we get:

\frac{21}{4} = p

Conclusion

In this article, we have found the value of p such that the given lines are perpendicular to each other. We used the concept of the slope of a line and the condition for perpendicularity to solve this problem. The value of p is \frac{21}{4}.

Example Problems

Find the value of p such that the lines $\frac{3-x}{5}=\frac{2y+6}{4p}=\frac{z-2}{-3}$ and $\frac{2+2x}{p}=\frac{y-2}{3}=\frac{z-3}{-1}$ are perpendicular to each other.
Find the value of p such that the lines $\frac{1-x}{2}=\frac{3y+9}{6p}=\frac{z-1}{-2}$ and $\frac{3+3x}{p}=\frac{y-3}{4}=\frac{z-2}{-2}$ are perpendicular to each other.

Solutions

The value of p is \frac{15}{2}.
The value of p is \frac{10}{3}.

Discussion

References

"Perpendicular Lines" by Math Open Reference
"Slope of a Line" by Math Is Fun
"Condition for Perpendicularity" by Khan Academy

Glossary

Perpendicular Lines: Two lines that intersect at a right angle.
Slope of a Line: A measure of how steep a line is.
Condition for Perpendicularity: A condition that two lines must satisfy in order to be perpendicular.

Index

Perpendicular Lines: 1, 2, 3
Slope of a Line: 1, 2, 3
Condition for Perpendicularity: 1, 2, 3

Introduction

In our previous article, we discussed how to find the value of p such that the given lines are perpendicular to each other. We used the concept of the slope of a line and the condition for perpendicularity to solve this problem. In this article, we will answer some frequently asked questions (FAQs) related to finding the value of p for perpendicular lines.

Q: What is the condition for perpendicularity?

A: The condition for perpendicularity is that the product of the slopes of two lines must be -1.

Q: How do I find the slope of a line?

A: To find the slope of a line, you can use the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Q: What is the formula for finding the value of p?

A: The formula for finding the value of p is: p = -1 / (m1 * m2), where m1 and m2 are the slopes of the two lines.

Q: Can I use the formula to find the value of p if the lines are not in the standard form?

A: Yes, you can use the formula to find the value of p even if the lines are not in the standard form. However, you need to rewrite the lines in the standard form first.

Q: What if the lines are parallel?

A: If the lines are parallel, then they have the same slope, and the product of their slopes is not -1. In this case, the lines are not perpendicular.

Q: Can I use the formula to find the value of p if the lines are not perpendicular?

A: No, you cannot use the formula to find the value of p if the lines are not perpendicular. The formula is only valid for perpendicular lines.

Q: What if I have a system of linear equations?

A: If you have a system of linear equations, you can use the formula to find the value of p by first solving the system of equations to find the slopes of the lines.

Q: Can I use the formula to find the value of p if the lines are not in the same plane?

A: No, you cannot use the formula to find the value of p if the lines are not in the same plane. The formula is only valid for lines in the same plane.

Q: What if I have a line in 3D space?

A: If you have a line in 3D space, you can use the formula to find the value of p by first finding the slope of the line in the x-y plane.

Q: Can I use the formula to find the value of p if the lines are not straight?

A: No, you cannot use the formula to find the value of p if the lines are not straight. The formula is only valid for straight lines.

Q: What if I have a curve?

A: If you have a curve, you cannot use the formula to find the value of p. The formula is only valid for straight lines.

Q: Can I use the formula to find the value of p if the lines are not in the same coordinate system?

A: No, you cannot use the formula to find the value of p if the lines are not in the same coordinate system. The formula is only valid for lines in the same coordinate system.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to finding the value of p for perpendicular lines. We hope that this article has been helpful in clarifying any doubts you may have had.

Example Problems

Find the value of p such that the lines $\frac{2-x}{3}=\frac{4y+8}{6p}=\frac{z-3}{-2}$ and $\frac{4+4x}{p}=\frac{y-4}{5}=\frac{z-5}{-1}$ are perpendicular to each other.
Find the value of p such that the lines $\frac{1-x}{2}=\frac{3y+9}{6p}=\frac{z-1}{-2}$ and $\frac{3+3x}{p}=\frac{y-3}{4}=\frac{z-2}{-2}$ are perpendicular to each other.

Solutions

The value of p is $\frac{15}{2}$ .
The value of p is $\frac{10}{3}$ .

Discussion

References

"Perpendicular Lines" by Math Open Reference
"Slope of a Line" by Math Is Fun
"Condition for Perpendicularity" by Khan Academy

Glossary

Perpendicular Lines: Two lines that intersect at a right angle.
Slope of a Line: A measure of how steep a line is.
Condition for Perpendicularity: A condition that two lines must satisfy in order to be perpendicular.

Index

Perpendicular Lines: 1, 2, 3
Slope of a Line: 1, 2, 3
Condition for Perpendicularity: 1, 2, 3