Problem 30 - Precalculus Mathematics for Calculus

Question

A description of a line is given. Find parametric equations for the line. The line parallel to the \(y\) -axis that crosses the \(x z\) -plane where \(x=-3\) and \(z=2\)

Step-by-Step Solution

Answer

The parametric equations are \( x = -3, \) \( y = t, \) and \( z = 2 \).

Step 1: Identify the point on the line

The line crosses the \( xz \)-plane at the point \( (x, y, z) = (-3, 0, 2) \). This is where the line intersects the plane \( y = 0 \).

Step 2: Determine the direction vector

Since the line is parallel to the \( y \)-axis, its direction vector will be \( \mathbf{d} = (0, 1, 0) \). This indicates that the movement along the line is only in the y-direction.

Step 3: Formulate parametric equations

Using the point \((-3, 0, 2)\) and the direction vector \((0, 1, 0)\), the parametric equations of the line can be written as: \[ x = -3, \]\[ y = t, \]\[ z = 2, \]where \( t \) is a parameter.

Key Concepts

Direction Vector

In geometry, the **direction vector** plays a crucial role when determining the orientation and direction of a line in space. Imagine a line that extends infinitely in both directions. The direction can be described using a vector, which is essentially an arrow pointing from one point on the line to another.
For a line parallel to the **y-axis**, the change occurs only along the y-dimension, meaning that the vector of movement must reflect this uniqueness. In our exercise, this is represented by the vector

• \( \mathbf{d} = (0, 1, 0) \)

This implies that while moving along the line, the value of y will change, but x and z remain constant. This simplifies the parametric representation since the direction vector aligns perfectly with just the change in y.
The direction vector not only tells us the path the line takes but also ensures that any calculations based on the line retain the intended directionality and integrity.

X Z-Plane

The **x z-plane** is a fundamental concept in three-dimensional geometry and physics. It represents a plane where the y-value remains constant, typically set to zero. When talking about intersections, such as where a line meets this plane, the coordinates

• must reflect no y-component \((x, 0, z)\)

Understanding how lines intersect with the x z-plane helps in visualizing spatial relationships and geometries.
In the given exercise, the line crosses the **x z-plane** at

• \((-3, 0, 2)\)

This indicates the exact point on the plane where the line exists. Lines can be imagined as either penetrating or lying entirely within a plane, but here, since the direction vector is inclined to the y-axis (and thus perpendicular to the x z-plane), the line only intersects at a particular point.

Parallel to Y-Axis

Lines **parallel to the y-axis** have unique properties in space. Such lines exhibit no inclination towards the x or z-axes. Essentially, they are vertical lines in the three-dimensional coordinate system, exhibiting motion or extension solely along the y-axis.
When determining parametric equations for such lines, you'll find that the x and z coordinates remain fixed, while the y coordinate changes. Using the line's point of intersection

• \((-3, 0, 2)\)

...with the x z-plane, and the knowledge that it is parallel to the y-axis, the parametric equations are elegantly simple:

• X remains constant at \(-3\),
• z stays put at \(2\),
• while y varies as the parameter \(t\).

This configuration—where only one variable changes to parameterize a system—demonstrates how being parallel to an axis simplifies both understanding and calculations related to lines in three-dimensional space.