Intersection Point: Line And Plane Explained

Hey guys! Let's dive into a cool math problem: figuring out where a line defined by parametric equations meets a plane. Specifically, we're looking at the point where the line given by x = 1 + t, y = 6 + 2t, and z = 2 + t intersects with the plane described by the equation x - y + z + 6 = 0. Sounds complicated? Don't worry, we'll break it down step by step to make it super easy to understand. This is a common type of problem in analytic geometry, and knowing how to solve it is a valuable skill.

Understanding the Problem: Line, Plane, and Intersection

First off, let's get our heads around what we're dealing with. We have a line in 3D space. The equations x = 1 + t, y = 6 + 2t, and z = 2 + t are called parametric equations, and they tell us the position of any point on the line in terms of a parameter, 't'. Think of 't' as a variable that changes, and as it changes, the point (x, y, z) moves along the line. Then, we have a plane. The equation x - y + z + 6 = 0 represents all the points (x, y, z) that lie on this flat surface. Our goal is to find the exact point where the line and the plane cross each other. That point, if it exists, is the intersection point. The main goal is to find the value of t and the corresponding (x, y, z) coordinates. We will use the parametric equation to make the computation easier to solve.

Now, to find this intersection point, we need to find the specific values of x, y, and z that satisfy both the line's equations and the plane's equation. It's like finding a needle in a haystack, but we've got a systematic approach to make it manageable. Remember, if the line and the plane don't intersect, there will be no solution. But in this case, we know there's an intersection, so we're good to go.

Essentially, the process involves substituting the expressions for x, y, and z from the line's parametric equations into the plane's equation. This will give us a single equation with only 't' as the variable. Solving for 't' gives us the value of the parameter at the intersection point. After finding 't', we can plug this value back into the parametric equations to find the coordinates (x, y, z) of the intersection point. So, let's get started and break it down further so that it can be easier to understand.

Solving for the Intersection Point

Okay, let's get down to business and actually solve this problem. Here's the play-by-play of how we find the intersection point. This is where we put our understanding of the concepts into action. This approach guarantees that we get to the correct solution.

Step 1: Substitution

First things first, we're going to use a technique called substitution. We have the equations for our line:

• x = 1 + t
• y = 6 + 2t
• z = 2 + t

And we have the equation for our plane: x - y + z + 6 = 0.

The idea is to substitute the x, y, and z values from the line equations into the plane equation. This way, we'll get an equation that only has 't' in it, which we can solve. Doing the substitution, the plane equation becomes: (1 + t) - (6 + 2t) + (2 + t) + 6 = 0.

Step 2: Simplify and Solve for t

Now that we've substituted, we need to simplify the equation and solve for 't'. Let's clean it up step by step:

1. First, remove the parentheses: 1 + t - 6 - 2t + 2 + t + 6 = 0.
2. Next, combine like terms. Notice that we have t - 2t + t, which simplifies to 0t (or just 0). Then, we have 1 - 6 + 2 + 6, which simplifies to 3. So, the equation becomes:
   3 = 0.

Wait a minute! This is unusual. We ended up with 3 = 0. What does this mean? It means there is no value of 't' that can satisfy this equation. It implies something about the original problem. This is a key step, so pay close attention. It is crucial to understand that math problems may not always have a solution. In the context of our problem, it indicates that the line and the plane do not intersect. They are parallel. There's no single point where they meet.

Step 3: Finding the Coordinates (if an intersection exists)

Since, in our case, the line and the plane don't intersect, we can't find the coordinates of an intersection point. If, however, we had found a value for 't', the next step would be to substitute that 't' back into the parametric equations of the line to find the x, y, and z coordinates of the intersection point. Let's pretend for a moment that we did find a value for t.

Let's say, just for illustration, that we found t = 1. We would then plug this value into our parametric equations:

• x = 1 + 1 = 2
• y = 6 + 2(1) = 8
• z = 2 + 1 = 3

This would give us the point (2, 8, 3). But remember, in our actual problem, the line and plane don't intersect, so this step isn't applicable. So remember, the final goal is to substitute the value of 't' in the coordinates and obtain the intersection point, but in this case, there isn't any.

Conclusion: No Intersection Point Found!

Alright, guys, so we've worked through the problem, and we've found that the line defined by the parametric equations x = 1 + t, y = 6 + 2t, and z = 2 + t does not intersect the plane given by the equation x - y + z + 6 = 0. This outcome tells us that the line is parallel to the plane. The key takeaway here is the process: we substituted the line equations into the plane equation, simplified, and solved for 't'. The fact that we ended up with a contradictory statement (3 = 0) told us that there's no solution. Remember, not every math problem has a solution. This is an important concept in math. Understanding this process will help you tackle similar problems with confidence. The parametric equations and linear algebra concepts are essential in many fields, including computer graphics, physics, and engineering. Keep practicing, and you'll get the hang of it!

Let's recap the critical steps:

1. Understand the Equations: Know what parametric equations and plane equations represent.
2. Substitution: Substitute the line equations into the plane equation.
3. Solve for t: Simplify and solve the resulting equation for 't'.
4. Interpret the Result: If you get a value for 't', plug it back into the line equations to find the intersection point. If you get a contradiction, the line and plane do not intersect.

Keep up the great work, and don't hesitate to practice more problems. Practice makes perfect, and with each problem, you'll become more confident in your mathematical abilities. Happy problem-solving!