Problem 30 - Algebra and Trigonometry

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

Parametric equations: \(x = -3\), \(y = t\), \(z = 2\).

Step 1: Identify Line Characteristics

The line is parallel to the y-axis. This indicates that its direction vector should only have a component in the y-direction and can be expressed as \(\mathbf{d} = (0, 1, 0)\).

Step 2: Determine a Point on the Line

The problem states that the line crosses the xz-plane at \(x = -3\) and \(z = 2\). Therefore, a point on the line can be represented as \(\mathbf{P} = (-3, 0, 2)\).

Step 3: Write the Parametric Equations

Using the point \((-3, 0, 2)\) and the direction vector \((0, 1, 0)\), we can express the parametric equations for the line as follows:\[x = -3, \y = t, \z = 2,\]where \(t\) is the parameter representing any real number.

Key Concepts

Line Parallel to the Y-Axis

A line parallel to the y-axis means that it extends indefinitely in the direction of the y-axis without deviating. This type of line runs vertically and has no horizontal component in either the x or z directions. Lines parallel to the y-axis are unique because they are entirely determined by their constant x and z values at any point along the line.

Here's why it's special:

• Only the y-coordinate changes — all other coordinates remain constant.
• Graphically, it appears as a vertical line when looking at the x-y or z-y plane.

In our given problem, the line intersects the xz-plane at a specific point while maintaining a direction exclusively along the y-axis. This characteristic simplifies the determination of its parametric equations significantly.

Direction Vector

The direction vector of a line is crucial in determining how the line extends in space. For a line parallel to the y-axis, the direction vector will have a component in the direction of the y-axis, and zero components in the x and z directions. In essence, the direction vector points the line's direction.

Understanding the direction vector:

• A vector \((0, 1, 0)\) means only motion along the y-axis.
• The vector's components tell us how far the line moves in each direction per unit of the parameter.

In the context of this exercise, since the line is specifically parallel to the y-axis, the direction vector is logically \((0, 1, 0)\). This vector indicates that as the parameter \(t\) varies, the line moves vertically along the y-axis without shifting in the x or z directions.

Point on the Line

A point on the line is essential for establishing the line's position in the 3D coordinate system. In our problem, the line crosses the xz-plane at \(x = -3\) and \(z = 2\), which gives us a specific location through which the line passes.

This point, \((-3, 0, 2)\), acts as an anchor:

• Indicates a specific location the line definitely passes through.
• The x and z coordinates are constant along the line since it is parallel to the y-axis.

In parametric equations, these coordinates are utilized to embed the line within the space along with the direction vector. The point provides the starting position, from which variations occur solely along the direction vector as the parameter \(t\) changes. Hence, knowing this point ensures that the line is correctly situated.