Problem 11 - University Calculus: Early Transcendentals

Question

Find parametric equations for the lines. The \(x\) -axis

Step-by-Step Solution

Answer

Parametric equations: \( x = t \), \( y = 0 \), \( z = 0 \).

Step 1: Understanding the Problem

We need to find parametric equations for the line that lies on the x-axis. The x-axis is a horizontal line at y = 0 and it extends infinitely in both positive and negative directions of the x-plane.

Step 2: Choosing a Point on the x-axis

Any point on the x-axis can be represented as (x, 0, 0). A simple point to start with is the origin, (0,0,0), which lies on the x-axis.

Step 3: Determining the Direction Vector

The x-axis stretches in both directions along the x-coordinate. The direction vector for the x-axis can be given by (1, 0, 0), indicating movement along the x-axis with no displacement in the y or z direction.

Step 4: Writing the Parametric Equations

Using the point (0,0,0) and the direction vector (1, 0, 0), the parametric equations of the line on the x-axis can be expressed as: \[ x = t \] \[ y = 0 \] \[ z = 0 \] where \( t \) is a parameter.

Key Concepts

Direction Vector

A direction vector is crucial in defining the direction of a line in space. For a line, the direction vector provides information on how the line extends along a coordinate axis. In this particular exercise, the direction vector is selected for the x-axis.

• Direction vectors consist of coordinates that show movement along each axis.
• For the x-axis, the direction vector is (1, 0, 0).
• This implies movement along the x-axis, with no change in the y or z directions.

Direction vectors help in writing parametric equations by determining how far along the line the parameter progresses.

X-Axis Lines

Lines along the x-axis are special due to their simplicity and alignment with a principal coordinate axis. These lines are horizontal within a 3D space and extend infinitely left and right relative to the origin.

• The x-axis is defined by the equation y = 0 and z = 0.
• The origin point (0, 0, 0) is a natural point of reference on this line.
• All points on the line have a y-coordinate and z-coordinate equal to zero.

Understanding x-axis lines makes it easier to tackle coordinate geometry problems, as they serve as a foundational concept in mathematics.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry through algebra and coordinates. It allows us to describe geometric shapes, like lines and curves, using numbers and equations.

• It involves the use of coordinate axes to describe positions of points.
• Parametric equations are one way of expressing lines and curves in coordinate geometry.
• Finding parametric equations involves using a starting point and a direction vector.

By applying coordinate geometry, complex geometric figures can be analyzed numerically, which enhances understanding and problem-solving abilities.

Vector Mathematics

Vector mathematics provides techniques for depicting and solving geometric problems using vectors. Vectors are quantities that have both magnitude and direction, making them ideal for representing lines and directions in space.

• Vectors can represent direction and magnitude, useful for plotting lines.
• In 3D space, vectors are typically expressed in terms of i, j, k unit vectors along the x, y, and z axes.
• By combining vectors with scalars, parametric equations can represent lines.

Vector mathematics not only simplifies complex spatial problems, but also provides a visual and numerical means to explore the nature of lines and other geometric entities.