Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the spiral \(\mathbf{r}=t \cos t \mathbf{i}+t \sin t \mathbf{j}+t \mathbf{k}\) lies on the circular cone \(x^{2}+y^{2}-z^{2}=0\). On what surface does the spiral \(\mathbf{r}=3 t \cos t \mathbf{i}+t \sin t \mathbf{j}+t \mathbf{k}\) lie?

Step-by-Step Solution

Answer

The spiral lies on the cone \(x^2 + y^2 - z^2 = 0\). The second spiral does not lie on such a simple surface.

Step 1: Parameterize the Spiral

The parametrized form of the spiral is given by \( \mathbf{r}=t \cos t \mathbf{i} + t \sin t \mathbf{j} + t \mathbf{k} \). This means the components in the \(x\), \(y\), and \(z\) directions are \((x, y, z) = (t \cos t, t \sin t, t) \).

Step 2: Substitute into the Cone Equation

The equation of the cone is \(x^2 + y^2 - z^2 = 0\). Substitute \(x = t \cos t\), \(y = t \sin t\), and \(z = t\) into the cone equation:\[(t \cos t)^2 + (t \sin t)^2 - t^2 = 0.\]

Step 3: Simplify the Expression

With the substitution made, the expression becomes:\[t^2 \cos^2 t + t^2 \sin^2 t - t^2 = 0.\]Apply the Pythagorean identity \(\cos^2 t + \sin^2 t = 1\):\[t^2 (\cos^2 t + \sin^2 t) - t^2 = 0 \t^2 (1) - t^2 = 0 \0 = 0.\]This proves the points lie on the cone.

Step 4: Analyze the New Spiral

The new spiral is \( \mathbf{r}=3t\cos t\mathbf{i}+t\sin t\mathbf{j}+t\mathbf{k} \). Identify the vector components: \((x, y, z) = (3t \cos t, t \sin t, t)\).

Step 5: Substitute into the Surface Equation

For this new spiral, let's substitute into a similar surface equation as before. However, identify the relation:\(x^2 + y^2 - z^2 = 9t^2 \cos^2 t + t^2 \sin^2 t - t^2\).

Step 6: Simplify the Expression

Calculate as follows:\[9t^2 \cos^2 t + t^2 \sin^2 t - t^2 \]This simplifies to:\[t^2 (9\cos^2 t + \sin^2 t - 1).\]This expression does not equal zero for all \(t\). Thus, it does not strictly conform to a specific surface equation like the previous one.

Key Concepts

Parametric Equations

Parametric equations are a way to describe a curve or a surface using parameters. This technique allows us to express spatial curves by defining each point's coordinates in terms of a common parameter, often denoted as \(t\). In the exercise above, the spiral is parametrized with \( \mathbf{r}=t \cos t \mathbf{i} + t \sin t \mathbf{j} + t \mathbf{k} \), where \(t\) is the parameter:

• \(x = t \cos t\)
• \(y = t \sin t\)
• \(z = t\)

Each coordinate depends on \(t\), which changes as you move along the spiral. This flexible representation makes it easier to manipulate and visualize complex curves, especially in 3D space. You may think of \(t\) as time, and as time progresses, you trace out the path of the spiral.

Conic Sections

Conic sections are the curves formed by the intersection of a plane with a cone. These include ellipses, parabolas, hyperbolas, and circles. In the exercise, the spiral is analyzed in relation to a circular cone. The cone equation utilized is:\[x^2 + y^2 - z^2 = 0\]This equation represents a double cone with its axis aligned with the \(z\)-axis. It describes the location of the spiral as it lies on the cone. By substituting the spiral's parametric equations into the cone's equation, we show:\[t^2 (\cos^2 t + \sin^2 t) - t^2 = 0\]Confirming the spiral lies entirely on the cone. This is a beautiful example of how parametric equations can be used to relate to conic sections in a 3D context.

Pythagorean Identity

The Pythagorean identity is a fundamental mathematical principle stating that for any angle \(t\):\[\cos^2 t + \sin^2 t = 1\]This identity is instrumental in simplifying equations involving trigonometric functions, like in our exercise. When substituting the parametric equations into the cone's equation, the step utilizes the identity to verify the spiral's path:\[t^2 (\cos^2 t + \sin^2 t) - t^2 = t^2 (1) - t^2 = 0\]Here, the Pythagorean identity transforms the equation from a trigonometric expression into a simple algebraic identity. This transformation confirms that the original spiral lies on the given cone. The identity itself is crucial because it holds true for any angle \(t\), simplifying the verification process significantly.

3D Coordinate Geometry

3D coordinate geometry allows us to explore and analyze geometric figures within three-dimensional space, using coordinates \((x, y, z)\). It extends the concepts of 2D geometry, enabling the study of shapes like spheres, planes, and more complex forms such as the spiral in our exercise.

• The original spiral is represented by three coordinates depending on \(t\): \((t \cos t, t \sin t, t)\).
• In 3D space, these coordinates map each point of the spiral as it ascends, demonstrating its path on the cone.

The task involves checking how changes to these parametric representations affect their placement relative to geometric surfaces like cones. By observing how a change in one component of \( \mathbf{r}\) alters its relationship with its surrounding surface, the second spiral illustrates different interactions within the 3D space:For the new spiral \( \mathbf{r}=3t\cos t\mathbf{i}+t\sin t\mathbf{j}+t\mathbf{k} \), the new spiral doesn't conform to the same cone equation due to the altered parameterization. This shows how geometry can adapt in a three-dimensional environment and the importance of precise parameters to fit specific geometric definitions.