Finding The Intersection: Line And Plane In Math
Hey guys! Let's dive into a cool math problem where we'll figure out where a line and a plane decide to meet. Specifically, we're talking about finding the intersection point of a line 'r' and a plane 'Ï€'. The line 'r' is defined by the parametric equation r(t) = (2 + t, 3 - 2t, 1 + 3t), and the plane 'Ï€' is defined by the equation 2x - y + z = 5. Sounds a bit complex, right? But don't worry, we'll break it down step by step and make it super understandable. This is a classic problem in linear algebra, and understanding how to solve it is key to grasping more advanced concepts. Think of it like this: you're trying to find the one single point in space that belongs to both the line and the plane. It's like finding a secret meeting spot! Ready? Let's get started.
We will use a detailed and easy-to-understand approach to guide you through the process, covering the essential steps required to solve this problem. First, we will learn about the parametric equation of a line and how it represents a line in three-dimensional space. Then, we will explore the equation of a plane and how it describes a flat surface. Finally, we will combine our knowledge of lines and planes to find the point where they intersect. This process involves substituting the parametric equations of the line into the equation of the plane, solving for the parameter 't', and substituting the value of 't' back into the parametric equations of the line to find the coordinates of the intersection point. So, buckle up, because we're about to put on our mathematical detective hats and solve this mystery! Understanding this will help you with a variety of mathematical concepts, and will let you better understand more advanced topics in the world of mathematics, such as vector spaces and more. The world of math is filled with puzzles, and this one is a fun one to try.
Understanding the Parametric Equation of a Line
Alright, let's start with the basics. The parametric equation of a line is just a fancy way of describing every single point that lies on a line in space. In our case, the line 'r' is defined by r(t) = (2 + t, 3 - 2t, 1 + 3t). What does this even mean, right? Well, think of 't' as a parameter. It's a variable that can take on any real value. For every value of 't', you get a different point on the line. For example, if t = 0, then r(0) = (2, 3, 1). If t = 1, then r(1) = (3, 1, 4). If t = -1, then r(-1) = (1, 5, -2). Each of these points, and infinitely many others, all lie on the same line. The parametric equation essentially tells us the coordinates (x, y, z) of any point on the line in terms of this parameter 't'.
Let's break down the equation r(t) = (2 + t, 3 - 2t, 1 + 3t) a bit further. It's essentially saying:
- x = 2 + t
- y = 3 - 2t
- z = 1 + 3t
These are the coordinate equations of the line. They tell you exactly how the x, y, and z coordinates change as the parameter 't' changes. The '2', '3', and '1' in the equation represent a specific point on the line, and the coefficients of 't' (1, -2, and 3) determine the direction vector of the line. The direction vector shows you which way the line is going in space. So, the parametric equation gives us a complete description of the line, allowing us to pinpoint any point along its path by simply plugging in the right value for 't'. Pretty cool, huh? The direction vector is what determines the slope of the line in 3D space. The direction vector is very important, because it allows us to know which direction the line is moving in the space that it is on.
Decoding the Equation of a Plane
Now, let's switch gears and talk about planes. A plane is a flat, two-dimensional surface that extends infinitely in all directions. The equation of a plane is typically written in the form Ax + By + Cz = D, where A, B, C, and D are constants. This equation represents all the points (x, y, z) that lie on the plane. In our problem, the plane 'Ï€' is defined by the equation 2x - y + z = 5. This equation is the same as saying that, for any point (x, y, z) on the plane, if you multiply x by 2, subtract y, and add z, you should always get 5. The equation 2x - y + z = 5 defines a specific plane in 3D space. Each of the coefficients of x, y, and z (2, -1, and 1, respectively) are important. These coefficients are used to calculate what's called the normal vector. The normal vector is a vector that is perpendicular to the plane.
So, if we have the equation 2x - y + z = 5, the normal vector to the plane is (2, -1, 1). This is a crucial concept. The normal vector helps to determine the orientation of the plane in space. The constant D (in this case, 5) determines how far the plane is from the origin. All points (x, y, z) that satisfy this equation are located on the plane. Visualizing a plane in 3D can be tricky, but imagine it as a flat surface extending indefinitely in all directions. When working with planes, it's really helpful to know how to move between different ways to represent the planes. This can help solve complex problems with more ease. The ability to switch equations from point-normal to normal form is a skill that will help you solve problems. Keep in mind that understanding how these equations work is essential when you want to solve math problems.
Finding the Intersection Point: The Grand Finale
Now comes the fun part: finding the intersection point. This is where the line and the plane meet. The strategy is to substitute the parametric equations of the line into the equation of the plane. Remember, our parametric equations of the line are:
- x = 2 + t
- y = 3 - 2t
- z = 1 + 3t
And the equation of the plane is 2x - y + z = 5. Let's substitute the values of x, y, and z from the line's equations into the plane's equation. This gives us:
2(2 + t) - (3 - 2t) + (1 + 3t) = 5
Now, we just need to solve for 't'. Let's simplify the equation:
4 + 2t - 3 + 2t + 1 + 3t = 5
Combining like terms, we get:
7t + 2 = 5
Subtracting 2 from both sides:
7t = 3
Dividing both sides by 7:
t = 3/7
So, we found that t = 3/7. This value of 't' is super important. It tells us exactly where the line and plane intersect. Now, we'll substitute this value of 't' back into the parametric equations of the line to find the coordinates of the intersection point. So, the value of 't' is exactly the value that will give us the intersection point in space! This can be a tricky concept, so make sure you understand this well!
Let's find the x, y, and z coordinates:
- x = 2 + t = 2 + (3/7) = 17/7
- y = 3 - 2t = 3 - 2(3/7) = 15/7
- z = 1 + 3t = 1 + 3(3/7) = 16/7
Therefore, the intersection point of the line 'r' and the plane 'Ï€' is (17/7, 15/7, 16/7). Congrats, we have solved the problem! We have successfully determined where a line and a plane meet. To ensure this, you can always test your findings by redoing the process, or inputting the information into a graphing calculator. This would allow you to visualize the whole process! This is a great example of how different mathematical concepts can be combined to solve problems. This intersection point (17/7, 15/7, 16/7) is the only point that is on both the line and the plane. Understanding how to find this intersection is crucial in many areas of mathematics and physics. Great work, we have made it!