Problem 26 - Advanced Engineering Mathematics

Question

Find parametric equations for the line through \((1,2,8)\) that is (a) parallel to the \(y\) -axis, and (b) perpendicular to the \(x y\) -plane.

Step-by-Step Solution

Answer

Line (a): \(x=1\), \(y=2+t\), \(z=8\). Line (b): \(x=1\), \(y=2\), \(z=8+t\).

Step 1: Understanding the Problem

The problem asks to find two different lines through a specific point, each with specific directional properties: one line parallel to the y-axis, and another perpendicular to the xy-plane.

Step 1: Line Parallel to the Y-Axis

For a line parallel to the y-axis, the direction vector should only have a component in the y-direction. Thus, a suitable direction vector is \( \langle 0, 1, 0 \rangle \). We use the point \((1, 2, 8)\) to write the parametric equations for the line as follows:\[x = 1y = 2 + tz = 8\]

Step 2: Line Perpendicular to the XY-Plane

A line perpendicular to the xy-plane will have its direction vector in the z-direction, i.e., \( \langle 0, 0, 1 \rangle \). Using the point \((1, 2, 8)\), the parametric equations for the line are:\[x = 1y = 2z = 8 + t\]

Key Concepts

Direction Vectors

Direction vectors are fundamental in describing lines in both 2D and 3D spaces. A direction vector indicates the direction in which a line extends. For any given line, this vector essentially points out how the line moves through space.

Consider a direction vector as an arrow. It has both magnitude and direction. In the context of parametric equations, a direction vector is crucial because it determines the rate of change along each axis. For instance, if you have a direction vector \(\langle 0, 1, 0 \rangle\), it means there's movement solely along the y-axis while keeping the x and z components constant.

When we mention vectors parallel to the y-axis, the direction vector will look like \(\langle 0, 1, 0 \rangle\). The absence of an x or z component signifies no deviation in those directions. For lines perpendicular to the xy-plane, the vector is \(\langle 0, 0, 1 \rangle\), indicating movement purely in the z-direction.

Understanding these vectors helps to form equations that correctly describe the lines in space you’re working with.

3D Coordinate Geometry

3D coordinate geometry extends the principles of 2D geometry into three dimensions, adding complexity as well as a deeper understanding of spatial relationships. Here, every point is defined using three coordinates: \((x, y, z)\).

The introduction of a z-coordinate allows us to represent points that deviate above or below the plane defined by the x and y coordinates. Imagine shifting a 2D grid (like a sheet of paper) into a 3D space (like a room); you now need a new value to describe how far off the paper a point is—this is the z-coordinate.

• In the 3D space, equations of lines can usually be represented in parametric form, involving a point on the line and a direction vector.
• The z-coordinate plays an essential role, particularly when considering planes such as the xy-plane, where z=0.
• Visualizing how lines and points relate in this expanded space boosts comprehension of problems involving orientation and intersection.

Parametric equations often serve to elegantly describe the behavior of a line, giving more context to its position and direction in 3D space.

Lines in Space

Lines in space, especially within 3D geometry, are represented through mathematical forms that consider all three dimensions: x, y, and z. These lines can be infinitely extended in one direction, described by a set of parametric equations.

Parametric equations for lines are defined by a point on the line and a direction vector. For a line through point \((x_0, y_0, z_0)\) with direction vector \(\langle a, b, c \rangle\), the parametric equations are:\[x = x_0 + at \y = y_0 + bt \z = z_0 + ct \]Here, "\(t\)" is the parameter that changes, tracing out the line as it varies.

Lines parallel to axes have simplified direction vectors, such as \(\langle 1, 0, 0 \rangle\) for x-axis parallel lines. If a direction vector has a zero in a component, the line does not change in that dimension.

• For example, lines parallel to the y-axis are given by equations such as \[x = 1\], \[y = 2 + t\] and \[z = 8\].
• Lines perpendicular to the xy-plane are expressed as \[x = 1\], \[y = 2\], and \[z = 8 + t\].

Recognizing these patterns helps in visualizing how the lines navigate through space and provides insight into their relations with various planes and vectors.