How to Prove Two Lines Intersect in 3D Geometry: A Comprehensive Guide

Understanding the concept of intersecting lines in three-dimensional (3D) geometry is fundamental in various branches of mathematics and applied sciences. This article provides a detailed guide on proving intersection between two lines in 3D space. We will break down the process into manageable steps and include an example to clarify the procedure.

Step 1: Represent the Lines Using Parametric Equations

Before we delve into the steps for determining intersection, let's first represent the two lines using parametric equations. Parametric equations allow us to describe the position of a point on the line as a function of a parameter.

Consider two lines, Line 1 and Line 2.

Line 1:

mathbf{r_1}t mathbf{p_1} t mathbf{d_1}

Here, mathbf{p_1} is a point on Line 1, mathbf{d_1} is the direction vector of Line 1, and t is the parameter.

Line 2:

mathbf{r_2}s mathbf{p_2} s mathbf{d_2}

Similarly, mathbf{p_2} is a point on Line 2, mathbf{d_2} is the direction vector of Line 2, and s is the parameter.

Step 2: Set the Equations Equal to Find the Intersection Point

To find the intersection, we set the parametric equations of both lines equal to each other:

mathbf{p_1} t mathbf{d_1} mathbf{p_2} s mathbf{d_2}

This results in a system of three linear equations:

start{align} p_{1x} t d_{1x} p_{2x} s d_{2x} p_{1y} t d_{1y} p_{2y} s d_{2y} p_{1z} t d_{1z} p_{2z} s d_{2z} end{align}

Where mathbf{p_1} [p_{1x}, p_{1y}, p_{1z}], mathbf{p_2} [p_{2x}, p_{2y}, p_{2z}], mathbf{d_1} [d_{1x}, d_{1y}, d_{1z}], and mathbf{d_2} [d_{2x}, d_{2y}, d_{2z}].

Step 3: Solve the System of Equations

Now, we need to solve the system of equations for the parameters t and s. Here are the steps involved:

Express s in terms of t or vice versa for one of the equations.

Substitute the expression into one of the other equations to find a single variable.

Solve for both t and s.

If a solution exists, it means the lines intersect at the point given by substituting t or s back into the respective line equation.

Step 4: Check for Consistency

If we find values for t and s that satisfy all three equations simultaneously, then the lines intersect at the point. If no such values exist, the lines do not intersect. In this case, you may want to consider whether the lines are skew, meaning they do not intersect and are not parallel.

Step 5: Conclusion

Here is an example to illustrate the process:

Example:

Let's say we have two lines:

Line 1: mathbf{r_1}t [1, 2, 3] t [4, 5, 6] Line 2: mathbf{r_2}s [7, 8, 9] s [10, 11, 12]

We want to determine if these lines intersect. First, we set the equations equal:

start{align} 1 t 4 7 s 10 2 t 5 8 s 11 3 t 6 9 s 12 end{align}

Next, we solve these equations for t and s.

Solving the first equation for s in terms of t:

s frac{t - 3}{5}

Substituting into the second equation:

2 t 5 8 frac{(t - 3)11}{5}

After solving, we find that there is no solution, indicating that the lines do not intersect.

Alternatively, if we find a solution, we can substitute t or s back into the respective line equation to find the intersection point.

By following these steps, you can determine whether two lines intersect in 3D space and understand the underlying mathematical processes involved.