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 behind your answer.

Step-by-Step Solution

Answer

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

Step 1: Identify Coordinate Planes

The coordinate planes are the planes formed by setting one of the three coordinates to zero. These planes are the xy-plane (where z = 0), the xz-plane (where y = 0), and the yz-plane (where x = 0).

Step 2: Find Intersection with the XY-Plane

To find the intersection with the xy-plane, set the z-coordinate of the line to zero: \[ z = 3t = 0 \] This gives \( t = 0 \). Substitute \( t = 0 \) into the equations for x and y:\[ x = 1 + 2(0) = 1 \] \[ y = -1 - 0 = -1 \] Thus, the point of intersection with the xy-plane is \((1, -1, 0)\).

Step 3: Find Intersection with the XZ-Plane

To find the intersection with the xz-plane, set the y-coordinate of the line to zero: \[ y = -1 - t = 0 \] Solving for \( t \) gives \( t = -1 \). Substitute \( t = -1 \) into the equations for x and z:\[ x = 1 + 2(-1) = -1 \] \[ z = 3(-1) = -3 \] Thus, the point of intersection with the xz-plane is \((-1, 0, -3)\).

Step 4: Find Intersection with the YZ-Plane

To find the intersection with the yz-plane, set the x-coordinate of the line to zero: \[ x = 1 + 2t = 0 \] Solving for \( t \) gives \( t = -\frac{1}{2} \). Substitute \( t = -\frac{1}{2} \) into the equations for y and z:\[ y = -1 - \left(-\frac{1}{2}\right) = -\frac{1}{2} \] \[ z = 3\left(-\frac{1}{2}\right) = -\frac{3}{2} \] Thus, the point of intersection with the yz-plane is \( \left(0, -\frac{1}{2}, -\frac{3}{2}\right) \).

Key Concepts

Understanding Intersection

In mathematics, an intersection is a point or a set of points where two or more geometric entities meet. In the context of coordinate planes, intersections are where a line or a curve crosses one of the planes.
When dealing with a line defined by parametric equations, the goal is to determine where this line intersects each of the coordinate planes. Let's first understand the basic planes:

• XY-Plane: Defined by setting the z-coordinate to zero.
• XZ-Plane: Defined by setting the y-coordinate to zero.
• YZ-Plane: Defined by setting the x-coordinate to zero.

To find these intersections, we substitute for zero in the non-relevant coordinate and solve for the parameter. This parameter is then used to find the remaining coordinates on the intersection point.
This method helps visualize how lines interact with these planes, offering deeper insights into the geometry of three-dimensional space.

Decoding Line Equations

Line equations on a coordinate plane can come in various forms. For three-dimensional space, lines are often expressed parametrically. In parametric equations, each coordinate (x, y, and z) of a point on the line is expressed as a function of a single parameter, usually denoted as \( t \).
For instance, in the line equation from our problem:

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

These equations indicate how each coordinate changes as \( t \) changes.

• The term in front of the parameter \( t \) represents the direction vector of the line.
• The constant term shows a particular point through which the line passes when \( t = 0 \).

By understanding these terms, one can easily derive how the line behaves and intersects different planes.

Exploring Cartesian Coordinates

The Cartesian coordinate system is a system that uses coordinates to uniquely determine the position of a point in space. It comprises axes that intersect at a point called the origin, usually denoted as \( (0, 0, 0) \) in three-dimensional space.
In three dimensions, these axes are:

• The x-axis: Extends horizontally.
• The y-axis: Extends vertically.
• The z-axis: Extends into and out of the plane of x and y.

Points are defined as \( (x, y, z) \), where each value corresponds to the position along the respective axis.
Understanding how to plot and interpret points in this system is crucial for solving geometry problems, such as finding intersections with coordinate planes. It establishes a fundamental understanding of three-dimensional spatial relationships, simplifying complex spatial reasoning.