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 \(\operatorname{spiral} \mathbf{r}=3 t \cos t \mathbf{i}+t \sin t \mathbf{j}+t \mathbf{k}\) lie?

Short answer

The spiral \(\mathbf{r} = t\cos t \mathbf{i} + t\sin t \mathbf{j} + t \mathbf{k}\) lies on the cone \(x^2 + y^2 - z^2 = 0\), while the spiral \(\mathbf{r} = 3t\cos t \mathbf{i} + t\sin t \mathbf{j} + t \mathbf{k}\) lies on a hyperbolic surface.

Step-by-step solution

Parameterization of the Spiral

The given vector \[ \mathbf{r} = t \cos t \mathbf{i} + t \sin t \mathbf{j} + t \mathbf{k} \] is a parametric equation describing a spiral in 3D space. The components are:- \( x(t) = t \cos t \)- \( y(t) = t \sin t \)- \( z(t) = t \)

Substitute into Cone Equation

Substitute the components into the cone equation \(x^2 + y^2 - z^2 = 0\):\[(t \cos t)^2 + (t \sin t)^2 - t^2 = 0\]Simplifying:\[t^2 \cos^2 t + t^2 \sin^2 t - t^2 = t^2 (\cos^2 t + \sin^2 t) - t^2 = 0\]

Simplify the Equation

Using the identity \(\cos^2 t + \sin^2 t = 1\), the equation simplifies to:\[t^2 (1) - t^2 = 0 \]This simplifies further to:\[0 = 0\]Therefore, the spiral lies on the given cone.

Analyze Second Spiral Parameterization

For the spiral \(\mathbf{r} = 3t \cos t \mathbf{i} + t \sin t \mathbf{j} + t \mathbf{k}\), the components are:- \( x = 3t \cos t \)- \( y = t \sin t \)- \( z = t \)

Determine the Surface of Second Spiral

Substitute the new components into the cone-like equation form \(x^2 + y^2 - z^2 = C\):\[(3t \cos t)^2 + (t \sin t)^2 - t^2 \]Simplifying:\[9t^2 \cos^2 t + t^2 \sin^2 t - t^2 = t^2 (9 \cos^2 t + \sin^2 t - 1) \]Using the identity \(\cos^2 t + \sin^2 t = 1\), we have:\[9 \cos^2 t + \sin^2 t = 9(1-\sin^2 t) + \sin^2 t = 8 \sin^2 t + 9\]Thus, it reflects a hyperbolic surface rather than a cone.

Key concepts

Circular Cones

A circular cone is a three-dimensional geometric figure with a circular base that narrows smoothly to a point called the apex or vertex. Understanding the equation of a circular cone helps in analyzing and visualizing curves that lie on its surface. The standard equation for a right circular cone with its vertex at the origin is given by:
\[ x^2 + y^2 - z^2 = 0 \]
This equation describes all the points \((x, y, z)\) in space that form a conical surface. It is symmetric around the z-axis, which acts as the axis of the cone. The cone opens equally in both directions along the z-axis. This can help students imagine how a curve, like a spiral, might wrap around the cone as it extends upward or downward.

Parametric Equations

Parametric equations are a set of equations that express the coordinates of the points making up a geometric object as functions of a variable, known as a parameter. In 3D geometry, they are essential for representing curves and surfaces.
Given the parametric curve \( \mathbf{r} = t \cos t \mathbf{i} + t \sin t \mathbf{j} + t \mathbf{k} \), the equations for the coordinates are:

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

Each component is a function of the parameter \( t \), describing how the curve moves through three-dimensional space as \( t \) changes. Parametric representation is particularly useful for curves that can't be easily expressed by standard algebraic equations. Different values of the parameter \( t \) yield different points on the curve, making it dynamic and versatile.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every allowed value of the occurring variable. These identities are essential in simplifying expressions and solving equations related to trigonometric functions.
One of the most fundamental identities used in the simplification of the parametric spiral is:
\[ \cos^2 t + \sin^2 t = 1 \]
This identity is crucial because it allows for simplification in the equation of the cone, verifying whether the spiral lies on the cone. By substituting values from parametric equations into a surface equation and utilizing trigonometric identities, complex geometric problems can often be made simpler.
Trigonometric identities also help in converting between different trigonometric forms, finding angles, and working through calculus problems involving trigonometric functions. They are indispensable tools in any mathematician's or engineer's toolkit.