Problem 29 - Precalculus Mathematics for Calculus

Question

A description of a line is given. Find parametric equations for the line. The line perpendicular to the \(x z\) -plane that lcontains the point \((2,-1,5)\)

Step-by-Step Solution

Answer

Parametric equations: \( x = 2 \), \( y = -1 + t \), \( z = 5 \).

Step 1: Understand the line's relationship to the plane

The line is perpendicular to the \( xz \)-plane. The \( xz \)-plane is defined by \( y = 0 \), meaning that any line perpendicular to it must have a direction vector pointing straight along the \( y \)-axis.

Step 2: Find the Direction Vector

Since the line is perpendicular to the \( xz \)-plane, it must have a direction vector \( \mathbf{d} = (0, 1, 0) \), which indicates movement only in the \( y \)-direction.

Step 3: Identify the Point on the Line

The line passes through the point \((2, -1, 5)\). This point is used as the initial point on the line.

Step 4: Write the Parametric Equations

The parametric equations involve moving from the given point in the direction of the direction vector. Using \( t \) as the parameter, the equations are: \[\begin{align*}x(t) &= 2 \y(t) &= -1 + 1t \z(t) &= 5\end{align*}\] This creates the parametric equations \( x = 2 \), \( y = -1 + t \), and \( z = 5 \).

Key Concepts

Direction Vector

A direction vector is fundamental when dealing with parametric equations involving lines. Essentially, it provides the line with its orientation in three-dimensional space. Imagine it as the arrow indicating which way the line stretches. For a line perpendicular to a particular plane, the direction vector is aligned with the axis that is perpendicular to that plane. In our exercise, the line is perpendicular to the \( xz \)-plane. This plane lies flat when you think of it as running along the \( x \) and \( z \) axes, which leaves the \( y \) axis as the vertical or perpendicular direction. Thus, the direction vector, given by \( \mathbf{d} = (0, 1, 0) \), aligns solely along the \( y \)-axis. This indicates that the line's movement occurs only up and down the \( y \) axis, highlighting its perpendicular nature to the \( xz \)-plane.

• Direction vectors play a pivotal role in parametric equations.
• They give the line its direction and define its geometric behavior.
• For lines perpendicular to a plane, the direction vector aligns along the axis that is not involved in that plane.

Perpendicular Lines

Perpendicular lines create a distinctive "T" shape when they intersect. For three-dimensional coordinate systems, these lines add more complexity, intersecting planes rather than lines. When a line is described as perpendicular to a plane, it means it meets the plane at a right angle, or 90 degrees.A plane, like the \( xz \)-plane, extends infinitely in both the \( x \) and \( z \) directions, with \( y = 0 \) as a constant. This implies that a line perpendicular to this plane will travel in the \( y \)-direction, showing no movement in the \( x \) and \( z \) directions. Therefore, in such scenarios, the direction vector reflects this constraint, ensuring parallel movement along one axis only. On a more intuitive level, envision a flagpole standing upright on a flat field; the pole is perpendicular to the Earth's surface, similar to our line being perpendicular to the \( xz \)-plane.

• Perpendicularity implies a 90-degree intersection to the surface or plane in question.
• A line perpendicular to a plane only moves along the axis not included in that plane's definition (like the \( y \) axis for the \( xz \)-plane).

Coordinate Planes

In three-dimensional space, coordinate planes serve as essential reference surfaces. Each plane is named based on the axes it aligns with. There are three primary planes: the \( xy \)-plane, the \( xz \)-plane, and the \( yz \)-plane.Focusing on the \( xz \)-plane, it is achieved by setting the \( y \) coordinate to zero. This implies that it stretches infinitely along the \( x \) and \( z \) coordinates. Coordinate planes essentially help us visualize and pinpoint locations in 3D spaces. They assist in expressing geometrical problems, like the perpendicularity of lines to specific planes. For a line passing through or perpendicular to the \( xz \)-plane, such insights allow for accurate representation using parametric equations, where each plane and line's relationship can be succinctly depicted using coordinate contexts.

• The main coordinate planes are the \( xy \)-plane, \( xz \)-plane, and \( yz \)-plane.
• Planes extend infinitely within the dimensions defined by their axes.
• The \( xz \)-plane is defined as where \( y = 0 \), stretching along the \( x \) and \( z \) axes.