Problem 23 - Advanced Engineering Mathematics

Question

Find parametric equations for the line through \((6,4,-2)\) that is parallel to the line \(x / 2=(1-y) / 3=(z-5) / 6\).

Step-by-Step Solution

Answer

The parametric equations are \( x = 6 + 2t, y = 4 - 3t, z = -2 + 6t \).

Step 1: Understand the given line

The given line is presented in its symmetric form: \( \frac{x}{2} = \frac{1-y}{3} = \frac{z-5}{6} \). The symmetric form can be separated into three equations that describe the parameterization. By recognizing this, we can identify the direction vector of the line, which is \( \langle 2, -3, 6 \rangle \).

Step 2: Identify a point and direction vector

We need a direction vector that is parallel to the given line. From Step 1, we know the direction vector of the given line is \( \langle 2, -3, 6 \rangle \). We also have a point through which our line passes, \((6,4,-2)\).

Step 3: Write the parametric equations

Using the direction vector \( \langle 2, -3, 6 \rangle \) and the point \((6, 4, -2)\), the parametric equations of the line can be constructed. The parametric equations are: \[x = 6 + 2t\,y = 4 - 3t\,z = -2 + 6t\] where \( t \) is the parameter.

Key Concepts

Symmetric Form

The symmetric form of a line in three-dimensional space beautifully compacts three equations into a single expression. Imagine having these expressions: \( \frac{x}{a} = \frac{b-y}{d} = \frac{z-c}{f} \). This expression reveals the relationship between variables on the line.
The behavior of a line in space is connected to its symmetric form through:

• It shows proportional relationships among the coordinates.
• It hints at the direction of the line with the denominators \(a, d, f\).

For the example line \( \frac{x}{2} = \frac{1-y}{3} = \frac{z-5}{6} \):

• These values \(2, -3, 6\) act as the direction vector components.
• They indicate how each axis changes.

This form allows quick conversion to parametric form by equating each part to a parameter \(t\).

Direction Vector

The direction vector is the secret compass of a line in space. For a line, it shows how the line extends.
In the symmetric form, the direction vector is clear from the denominators: \( \langle a, d, f \rangle \) or specifically \( \langle 2, -3, 6 \rangle \) for our example. This vector:

• Describes the line's direction.
• Is unaffected by starting points.

Hence, when finding lines parallel in space:

• Using the direction vector ensures parallelism.
• Varying only the start point gives a new line.

For every unit of movement in \(x\), you move proportionately in \(y \, \text{and} \, z\).

Line Equations

Bringing lines to life in space involves writing their equations. Lines in 3D have favorable representations, like parametric equations. Given a direction vector \( \langle a, d, f \rangle \) and a specific point \((x_0, y_0, z_0)\):
The line's parametric equations become:
\[ x = x_0 + at, \, y = y_0 + dt, \, z = z_0 + ft \]
In our example, using point \((6, 4, -2)\) and direction vector \( \langle 2, -3, 6 \rangle \):
\[ x = 6 + 2t, \, y = 4 - 3t, \, z = -2 + 6t \]
Where \( t \) is a real number. This gives a powerful way to trace any point along the line with varying \( t \). Lines can be:

• Infinite with respect to the parameter \( t \).
• Customized with different starting points but consistent directions.

Vectors in Space

Vectors in space form the backbone of how we navigate through three dimensions. They represent more than just points; they convey *direction and magnitude*.
Key features include:

• Describing positions, directions, and displacements.
• Characterization with components like \( \langle x, y, z \rangle \).

In line equations, they specifically:

• Indicate the movement's direction using components.
• Without starting point restrictions, define infinite parallel lines.

If you visualize moving along a line, you're traveling along its vector. Vectors are versatile in describing linear motion, showing how parts move in relation to each other in space.