Problem 27 - Advanced Engineering Mathematics

Question

Show that the lines given by \(\mathbf{r}=t(1,1,1\rangle\) and \(\mathbf{r}=\langle 6,6,6\rangle+\) \(t(-3,-3,-3)\) are the same.

Step-by-Step Solution

Answer

Both lines are the same, as their parametric equations produce identical points for all values of \( t \).

Step 1: Identify Parametric Equations

First, express both line equations in parametric form. The first line is defined by \( \mathbf{r} = t\langle 1, 1, 1 \rangle \) which gives the parametric equations: \( x = t, y = t, z = t \). The second line is given by \( \mathbf{r} = \langle 6, 6, 6 \rangle + t\langle -3, -3, -3 \rangle \), leading to the equations: \( x = 6 - 3t, y = 6 - 3t, z = 6 - 3t \).

Step 2: Equate Parametric Equations

Set the corresponding parametric equations of both lines equal to identify transformations. For the x-components: \( t = 6 - 3t \). Solving gives \( 4t = 6 \) or \( t = \frac{3}{2} \).

Step 3: Substitute Back to Verify

Substitute \( t = \frac{3}{2} \) into the parametric form of the first line, checking if the same point is reached as on the second line. The first line yields \( x = 1.5, y = 1.5, z = 1.5 \). Likewise, substituting \( t = \frac{3}{2} \) into \( 6 - 3t \) also gives \( x = 1.5, y = 1.5, z = 1.5 \) for the second line.

Step 4: Evaluate Consistency of Points

Both lines coincide at all points because substituting any \( t \) yields corresponding points; \( t = \frac{3}{2} \) particularly confirms intersection and identical trajectories. Thus, both lines represent the same geometric entity.

Key Concepts

Parametric Equations

In vector calculus, parametric equations are a convenient way to describe lines and curves in space.They use a parameter, often denoted by \( t \), to express coordinates.For example, a line can be represented as \( \mathbf{r} = \langle x(t), y(t), z(t) \rangle \).
This means each coordinate \( x, y, \text{and} z \) is a function of \( t \).
This is useful as it allows us to easily calculate points on the line by simply plugging in different values of \( t \).In our exercise, we start with two parametric equations corresponding to two lines.For the first line, \( \mathbf{r} = t\langle 1, 1, 1 \rangle \);
this gives parametric equations: \( x = t, y = t, z = t \).For the second line, \( \mathbf{r} = \langle 6, 6, 6 \rangle + t\langle -3, -3, -3 \rangle \);which results in: \( x = 6 - 3t, y = 6 - 3t, z = 6 - 3t \).
By equating these, we can see how they relate through transformations.
This step is crucial to show both lines represent the same geometric entity.

Line Equation

A line equation in vector calculus is another representation of a line besides parametric forms.It usually takes the form of \( \mathbf{r} = \mathbf{a} + t\mathbf{b} \),where \( \mathbf{a} \) is a specific point on the line and \( \mathbf{b} \) is the direction vector.
For any given \( t \), a point on the line can be found.This form is powerful in determining various properties of lines such as direction, distance between points, and intersections.In the example from the exercise, the first line's equation is \( \mathbf{r} = t\langle 1, 1, 1 \rangle \),with the direction vector \( \langle 1, 1, 1 \rangle \) starting from the origin.The second line \( \mathbf{r} = \langle 6, 6, 6 \rangle + t\langle -3, -3, -3 \rangle \)shifts the entire line through translations and direction vectors.
By showing that these two line equations produce identical parametric results at any \( t \),we conclude that they are essentially the same line, as verified by equating their equations.

Geometric Transformations

Geometric transformations involve manipulating the position, size, or orientation of a shape or line.In our context, it refers to how parametric line equations can transform one line into another by shifting it or changing its direction.This is achieved by adjusting the position vector and the direction vector in a line equation.In our exercise, the transformation is evident when comparing the two lines.The line \( \mathbf{r}=t\langle 1, 1, 1 \rangle \) is simply a scaling line starting at the origin.The line \( \mathbf{r}=\langle 6, 6, 6 \rangle + t\langle -3, -3, -3 \rangle \) undergoes a geometric transformation using:

• A translation by the vector \( \langle 6, 6, 6 \rangle \).
• A change in direction scales the vectors by \( -3 \).

Ultimately, these transformations modify the line's trajectory, but equivalently in the context of the given problem.Thus, by understanding geometric transformations, we can demonstrate how two lines appear different but are essentially representations of the same line.