3D Line Equation: Find The Equation Passing Through A & B

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Hey guys! Let's dive into a super interesting problem today: finding the equation of a line in 3D space. Specifically, we're tackling the question: What is the equation of the line in 3D space that passes through points A(2, 4, -1) and B(5, 0, 7)? This is a classic problem in spatial geometry, and I'm going to break it down step-by-step so it's crystal clear.

Understanding 3D Lines

Before we jump into the solution, it's crucial to understand how lines are represented in three-dimensional space. Unlike 2D lines, which can be defined simply by a slope and y-intercept, 3D lines require a bit more finesse. We typically describe a 3D line using either a parametric equation or a symmetric equation. Both representations rely on the concept of a direction vector.

  • Direction Vector: A direction vector, often denoted as v, indicates the line's orientation in space. It's a vector parallel to the line. We find it by subtracting the coordinates of two points on the line. In our case, points A and B will help us define this vector.
  • Parametric Equation: The parametric equation represents the line as a set of equations, one for each coordinate (x, y, z), in terms of a parameter, usually denoted as t. These equations express the coordinates of any point on the line as a function of t.
  • Symmetric Equation: The symmetric equation is derived from the parametric equations by solving each parametric equation for t and setting them equal to each other. This form provides a more concise representation, but it's not always applicable if one of the direction vector components is zero.

Step-by-Step Solution

Okay, let's get our hands dirty and solve the problem! Here’s how we can find the equation of the line passing through A(2, 4, -1) and B(5, 0, 7).

1. Find the Direction Vector

First things first, we need to determine the direction vector v of the line. This vector points along the line and can be found by subtracting the coordinates of point A from the coordinates of point B:

v = B - A = (5 - 2, 0 - 4, 7 - (-1)) = (3, -4, 8)

So, our direction vector v is (3, -4, 8). This vector tells us the line's orientation in 3D space.

2. Form the Parametric Equations

Now that we have the direction vector, we can write the parametric equations of the line. The general form of the parametric equations for a line passing through a point P(x₀, y₀, z₀) with direction vector v = (a, b, c) is:

  • x = x₀ + at
  • y = y₀ + bt
  • z = z₀ + ct

We can use either point A or point B as our reference point P. Let’s use point A(2, 4, -1). Plugging in the values, we get:

  • x = 2 + 3t
  • y = 4 - 4t
  • z = -1 + 8t

These are the parametric equations of the line! They describe every point on the line as t varies over all real numbers. Pretty cool, right?

3. Form the Symmetric Equations (If Possible)

To find the symmetric equations, we solve each parametric equation for t:

  • t = (x - 2) / 3
  • t = (y - 4) / -4
  • t = (z + 1) / 8

Now, we set these expressions for t equal to each other. This gives us the symmetric equations of the line:

(x - 2) / 3 = (y - 4) / -4 = (z + 1) / 8

The symmetric equations provide a compact representation of the line. However, if any component of the direction vector were zero, we wouldn't be able to use this form for that particular coordinate.

4. Verification (Optional but Recommended)

To make sure we haven't made any silly mistakes, it's always a good idea to verify our equations. We can do this by plugging in the coordinates of point B(5, 0, 7) into our parametric or symmetric equations and checking if they hold true.

Using the parametric equations:

  • 5 = 2 + 3t => t = 1
  • 0 = 4 - 4t => t = 1
  • 7 = -1 + 8t => t = 1

Since we get the same value of t (t = 1) for all three equations, point B lies on the line, which confirms our solution!

Different Representations

It's worth noting that there are multiple ways to represent the same line. We could have used point B instead of point A to form the parametric equations, and we would have arrived at a different set of equations that still represent the same line. Similarly, multiplying the direction vector by a scalar would also result in a different set of equations, but they would still describe the same line in 3D space.

Importance of Direction Vectors

The direction vector plays a pivotal role in defining the line. It provides the orientation and “slope” of the line in 3D. Think of it like this: if you're walking along the line, the direction vector tells you which way to go. The magnitude of the direction vector doesn't matter; only its direction is important.

Common Mistakes to Avoid

When tackling these problems, here are a few common pitfalls to watch out for:

  • Incorrect Direction Vector: Make sure you subtract the coordinates in the correct order (B - A, not A - B). The order matters because it determines the direction of the vector.
  • Algebra Errors: Be careful with your arithmetic, especially when solving for t in the symmetric equations.
  • Forgetting the Parameter: Remember that the parametric equations involve a parameter (t), which can take any real value. Don't just plug in specific values for t without understanding the concept.
  • Incorrectly Applying Formulas: Ensure you're using the correct formulas for parametric and symmetric equations. Double-check the signs and the positions of the variables.

Real-World Applications

Understanding lines in 3D space isn't just an abstract mathematical concept. It has tons of real-world applications! Here are just a few examples:

  • Computer Graphics: 3D lines are fundamental in computer graphics for rendering objects, creating perspective, and simulating lighting.
  • Robotics: Robots use 3D lines to plan paths, navigate environments, and manipulate objects.
  • Navigation Systems: GPS and other navigation systems rely on 3D geometry to determine positions and routes.
  • Engineering: Engineers use 3D lines to design structures, model mechanical systems, and analyze spatial relationships.
  • Physics: In physics, 3D lines are used to represent trajectories of objects, model forces, and describe fields.

Practice Problems

To solidify your understanding, try tackling some practice problems! Here are a few ideas:

  1. Find the equation of the line passing through points C(1, -2, 3) and D(4, 5, -6).
  2. Determine if the point (7, -4, 15) lies on the line you found in the original problem.
  3. Find the equation of a line parallel to the line passing through A and B, but passing through the origin (0, 0, 0).

Working through these problems will help you become more comfortable with 3D line equations and the underlying concepts.

Conclusion

So, finding the equation of a line in 3D space might seem a bit daunting at first, but by breaking it down into steps, it becomes much more manageable. We learned how to find the direction vector, form the parametric and symmetric equations, and verify our results. Remember, practice makes perfect, so keep those problems coming! Understanding these concepts is super valuable, not just in math class, but also in various real-world applications. Keep exploring, guys, and happy problem-solving!