Problem 65 - Thomas Calculus

Question

Find the points in which the line \(x=1+2 t, y=-1-t\) \(z=3 t\) meets the coordinate planes. Describe the reasoning be- hind your answer.

Step-by-Step Solution

Answer

The line meets the xy-plane at (1, -1, 0), the yz-plane at (0, -1/2, -3/2), and the xz-plane at (-1, 0, -3).

Step 1: Introduction

The line is expressed in parametric form, given by: \[x = 1 + 2t,\]\[y = -1 - t,\]\[z = 3t.\]Coordinate planes are determined by setting one of the coordinates to zero. This means we check where the line intersects the three planes: xy-plane, yz-plane, and xz-plane.

Step 1: Intersection with the xy-plane

The xy-plane is defined by \(z = 0\). To find where the line intersects the xy-plane, set \(z = 3t = 0\). Solving for \(t\):\[t = 0.\]Substitute \(t = 0\) back into the parametric equations:- \(x = 1 + 2(0) = 1\)- \(y = -1 - 0 = -1\)- \(z = 3(0) = 0\)Thus, the intersection point is \((1, -1, 0)\).

Step 2: Intersection with the yz-plane

The yz-plane is defined by \(x = 0\). Set \(x = 1 + 2t = 0\) and solve for \(t\):\[1 + 2t = 0 \t = -\frac{1}{2}.\]Substitute \(t = -\frac{1}{2}\) back into the parametric equations:- \(x = 1 + 2(-\frac{1}{2}) = 0\)- \(y = -1 - (-\frac{1}{2}) = -\frac{1}{2}\)- \(z = 3(-\frac{1}{2}) = -\frac{3}{2}\)Thus, the intersection point is \((0, -\frac{1}{2}, -\frac{3}{2})\).

Step 3: Intersection with the xz-plane

The xz-plane is defined by \(y = 0\). Set \(y = -1 - t = 0\) and solve for \(t\):\[-1 - t = 0 \t = -1.\]Substitute \(t = -1\) back into the parametric equations:- \(x = 1 + 2(-1) = -1\)- \(y = -1 - (-1) = 0\)- \(z = 3(-1) = -3\)Thus, the intersection point is \((-1, 0, -3)\).

Key Concepts

Parametric Equations

Parametric equations are a powerful way to describe curves and surfaces, especially in 3D geometry. Unlike traditional ways of writing equations, the variables are expressed as functions of one or more parameters.
In our exercise, the line is defined by the parameter \(t\):

• \(x = 1 + 2t\)
• \(y = -1 - t\)
• \(z = 3t\)

These equations give us the exact location of any point on the line, by simply choosing a value for \(t\).
This format is particularly useful for identifying where a line intersects with planes. Just by substituting the appropriate conditions, we can easily find these intersections.

Coordinate Planes

Coordinate planes are essential in understanding spatial relationships in 3D geometry. These planes split the three-dimensional space into sections and are defined by setting one of the three coordinates (x, y, or z) to zero.
In this problem, you can think of the planes as flat sheets lying across space:

• The xy-plane occurs where \(z = 0\).
• The yz-plane occurs where \(x = 0\).
• The xz-plane occurs where \(y = 0\).

These definitions allow us to find where the line interacts with these planes, by setting the corresponding variables to zero in the line's parametric equations.

3D Geometry

3D geometry is the branch of mathematics that allows us to analyze and visualize objects in a three-dimensional space.
This involves looking at how lines, planes, and shapes can be positioned and how they intersect.
Understanding the relationship between lines and planes is crucial. A line can intersect a plane at a single point, be parallel, or even lie within the plane itself.
In this context, the line from our exercise intersects each of the coordinate planes at distinct points, calculated through the parametric form of the line.

Solving Equations

Solving equations is at the heart of finding intersections in geometry. Given the parametric form of a line, solving equations involves substituting the conditions for each plane into the line's equations.
For example:

• Intersection with the xy-plane: set \(z = 0\), solve \(3t = 0\).
• Intersection with the yz-plane: set \(x = 0\), solve \(1 + 2t = 0\).
• Intersection with the xz-plane: set \(y = 0\), solve \(-1 - t = 0\).

In each case, solving the equation gives the value of \(t\), which we plug back into the line's parametric equations.
This process reveals the coordinates of the intersection points.