The Condition for Two Lines to Intersect

Oct 13, 2025 by windailys.com 99 views

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Introduction

In this article, we will explore the concept of finding the value of $$b$$ and the point of intersection of two lines in three-dimensional space. We will use the concept of parametric equations and the condition for two lines to intersect.

The Condition for Two Lines to Intersect

Two lines in three-dimensional space intersect if and only if their direction vectors are not parallel. In other words, if the two lines have direction vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ , then the lines intersect if and only if $\mathbf{v}_1 \times \mathbf{v}_2 \neq \mathbf{0}$ .

The Direction Vectors of the Two Lines

The direction vector of the first line is $\mathbf{v}_1 = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}$ , and the direction vector of the second line is $\mathbf{v}_2 = \begin{pmatrix} 5 \\ 2 \\ 1 \end{pmatrix}$ .

The Condition for the Two Lines to Intersect

Since the two lines have direction vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ , we need to check if $\mathbf{v}_1 \times \mathbf{v}_2 \neq \mathbf{0}$ . We can calculate the cross product of $\mathbf{v}_1$ and $\mathbf{v}_2$ as follows:

\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 5 & 2 & 1 \end{vmatrix} = \mathbf{i} \begin{vmatrix} 3 & 4 \\ 2 & 1 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 2 & 3 \\ 5 & 2 \end{vmatrix}

= \mathbf{i} (3 - 8) - \mathbf{j} (2 - 20) + \mathbf{k} (4 - 15)

= -5\mathbf{i} + 18\mathbf{j} - 11\mathbf{k}

Since the cross product is not equal to $\mathbf{0}$ , the two lines intersect.

The Value of $$b$$

To find the value of $$b$$, we need to find the point of intersection of the two lines. We can do this by equating the parametric equations of the two lines.

The parametric equations of the first line are:

x = 1 + 2t

y = b + 3t

z = 3 + 4t

The parametric equations of the second line are:

x = 4 + 5s

y = 1 + 2s

z = s

We can equate the $$x$$, $$y$$, and $$z$$ components of the two lines to get the following system of equations:

1 + 2t = 4 + 5s

b + 3t = 1 + 2s

3 + 4t = s

We can solve this system of equations to find the values of $$t$$, $$s$$, and $$b$$.

Solving the System of Equations

We can start by solving the third equation for $$s$$:

s = 3 + 4t

Substituting this expression for $$s$$ into the first equation, we get:

1 + 2t = 4 + 5(3 + 4t)

Simplifying this equation, we get:

1 + 2t = 19 + 20t

Subtracting $$2t$$ from both sides, we get:

1 = 19 + 18t

Subtracting $$19$$ from both sides, we get:

-18 = 18t

Dividing both sides by $$18$$, we get:

t = -1

Substituting this value of $$t$$ into the expression for $$s$$, we get:

s = 3 + 4(-1)

s = -1

Substituting these values of $$t$$ and $$s$$ into the second equation, we get:

b + 3(-1) = 1 + 2(-1)

Simplifying this equation, we get:

b - 3 = -1

Adding $$3$$ to both sides, we get:

b = 2

The Point of Intersection

The point of intersection of the two lines is given by the parametric equations of the first line, with $$t = -1$$:

x = 1 + 2(-1)

y = b + 3(-1)

z = 3 + 4(-1)

Substituting the value of $$b$$, we get:

x = -1

y = 2 - 3

z = -1

The point of intersection is $$(x, y, z) = (-1, -1, -1)$$.

Conclusion

In this article, we found the value of $$b$$ and the point of intersection of two lines in three-dimensional space. We used the concept of parametric equations and the condition for two lines to intersect. We solved a system of equations to find the values of $$t$$, $$s$$, and $$b$$, and we found the point of intersection of the two lines.

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Q: What is the condition for two lines to intersect?

A: Two lines in three-dimensional space intersect if and only if their direction vectors are not parallel. In other words, if the two lines have direction vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ , then the lines intersect if and only if $\mathbf{v}_1 \times \mathbf{v}_2 \neq \mathbf{0}$ .

Q: How do we find the value of $$b$$?

A: To find the value of $$b$$, we need to find the point of intersection of the two lines. We can do this by equating the parametric equations of the two lines.

Q: What are the parametric equations of the two lines?

A: The parametric equations of the first line are:

x = 1 + 2t

y = b + 3t

z = 3 + 4t

The parametric equations of the second line are:

x = 4 + 5s

y = 1 + 2s

z = s

Q: How do we equate the parametric equations of the two lines?

A: We can equate the $$x$$, $$y$$, and $$z$$ components of the two lines to get the following system of equations:

1 + 2t = 4 + 5s

b + 3t = 1 + 2s

3 + 4t = s

Q: How do we solve the system of equations?

A: We can start by solving the third equation for $$s$$:

s = 3 + 4t

Substituting this expression for $$s$$ into the first equation, we get:

1 + 2t = 4 + 5(3 + 4t)

Simplifying this equation, we get:

1 + 2t = 19 + 20t

Subtracting $$2t$$ from both sides, we get:

1 = 19 + 18t

Subtracting $$19$$ from both sides, we get:

-18 = 18t

Dividing both sides by $$18$$, we get:

t = -1

Substituting this value of $$t$$ into the expression for $$s$$, we get:

s = 3 + 4(-1)

s = -1

Substituting these values of $$t$$ and $$s$$ into the second equation, we get:

b + 3(-1) = 1 + 2(-1)

Simplifying this equation, we get:

b - 3 = -1

Adding $$3$$ to both sides, we get:

b = 2

Q: What is the point of intersection of the two lines?

A: The point of intersection of the two lines is given by the parametric equations of the first line, with $$t = -1$$:

x = 1 + 2(-1)

y = b + 3(-1)

z = 3 + 4(-1)

Substituting the value of $$b$$, we get:

x = -1

y = 2 - 3

z = -1

The point of intersection is $$(x, y, z) = (-1, -1, -1)$$.

Q: What is the significance of finding the value of $$b$$ and the point of intersection of two lines?

A: Finding the value of $$b$$ and the point of intersection of two lines is important in various fields such as physics, engineering, and computer science. It can be used to model real-world problems and make predictions about the behavior of systems.

Q: How can we apply the concept of finding the value of $$b$$ and the point of intersection of two lines in real-world problems?

A: We can apply the concept of finding the value of $$b$$ and the point of intersection of two lines in various real-world problems such as:

Modeling the motion of objects in physics
Designing and optimizing systems in engineering
Developing algorithms and models in computer science

Q: What are some common challenges when finding the value of $$b$$ and the point of intersection of two lines?

A: Some common challenges when finding the value of $$b$$ and the point of intersection of two lines include:

Solving systems of equations
Handling degenerate cases
Dealing with complex and non-linear equations

Q: How can we overcome these challenges?

A: We can overcome these challenges by:

Using numerical methods and algorithms
Employing advanced mathematical techniques and tools
Developing and using software and programming languages to solve and visualize the problems.