Problem 12 - Advanced Engineering Mathematics

Question

Find the vector function that describes the curve \(C\) of intersection between the given surfaces. Sketch the curve \(C\). Use the indicated parameter. $$ x^{2}+y^{2}-z^{2}=1, y=2 x ; x=t $$

Step-by-Step Solution

Answer

The vector function is \( \mathbf{r}(t) = \langle t, 2t, \pm \sqrt{5t^2 - 1} \rangle \).

Step 1: Analyze the given surfaces

We have two surfaces: a hyperboloid given by the equation \( x^2 + y^2 - z^2 = 1 \) and a plane given by the equation \( y = 2x \). Our goal is to find the curve of intersection between these two surfaces.

Step 2: Substitute for the plane equation

Since the plane equation is \( y = 2x \), we substitute \( y = 2x \) into the hyperboloid equation. This gives us \( x^2 + (2x)^2 - z^2 = 1 \).

Step 3: Simplify the hyperboloid equation

Substituting \( y = 2x \) into the hyperboloid equation results in: \[ x^2 + 4x^2 - z^2 = 1 \]. Simplifying this, we have: \[ 5x^2 - z^2 = 1 \].

Step 4: Express variables in terms of the parameter

Given that \( x = t \), substitute \( x = t \) into the equation \( 5x^2 - z^2 = 1 \). We have:\[ 5t^2 - z^2 = 1 \]. This simplifies to \( z^2 = 5t^2 - 1 \), and therefore \( z = \pm \sqrt{5t^2 - 1} \).

Step 5: Write the vector function

The position vector function describing the curve \( C \) is \( \mathbf{r}(t) = \langle t, 2t, \pm \sqrt{5t^2 - 1} \rangle \). This expression encompasses both branches of the intersection curve in the given parameter \( t \).

Step 6: Sketch the curve

For a sketch, plot the vector function \( \mathbf{r}(t) \). It represents a curve having its projection as a straight line in the \( xy \)-plane (since \( y = 2x \)) and a hyperbolic component in the \( z \)-axis.

Key Concepts

Understanding Hyperboloids

A hyperboloid is an interesting and significant surface in the realm of three-dimensional geometry. The equation associated with a hyperboloid is generally of the form \(x^2 + y^2 - z^2 = 1\) or similar variants. It consists of two types: a hyperboloid of one sheet and a hyperboloid of two sheets.
In our context, the hyperboloid given by \(x^2 + y^2 - z^2 = 1\) is known as a hyperboloid of one sheet.

• Its key feature is that for any fixed value of \(z\), the cross-sections are circles.
• It resembles a cooling tower or a smooth hourglass shape.
• Hyperboloids have a unique characteristic of having straight lines that lie entirely on the surface.

These properties are why understanding hyperboloids is crucial when examining intersections with other surfaces.

Exploring the Curve of Intersection

The curve of intersection is where two surfaces meet in three-dimensional space. In this exercise, the surfaces are a hyperboloid and a plane.
This curve often reflects both shapes' characteristics and can be an intriguing geometrical entity.

• For the surfaces \(x^2 + y^2 - z^2 = 1\) and \(y = 2x\), the intersection is a curve where both equations hold true simultaneously.
• Substituting \(y = 2x\) into the hyperboloid simplifies it significantly, allowing us to describe the curve more easily.
• The resulting curve appears as a twisted path extending through space, maintaining the planar relationship \(y = 2x\).

Visualizing this curve involves recognizing the dynamic interaction between linear and curved geometries.

Understanding Vector Functions

Vector functions are vital in representing curves in space. They utilize vectors to map from a parameter (usually \(t\)) to points on the curve.

• A vector function, like \(\mathbf{r}(t) = \langle t, 2t, \pm \sqrt{5t^2 - 1} \rangle \), provides each point's coordinate on the curve.
• It allows us to express the curve parametrically using \(t\) as a simple input, yielding a range of coordinates that describe the path.
• This transformation helps identify specific points or segments of the curve by varying \(t\).

Vector functions simplify the representation of complex curves and are fundamental in the study of vector calculus.

Decoding Parametric Equations

Parametric equations define a curve by expressing each coordinate as a separate function of an independent parameter, usually \(t\).

• In the vector function \(\mathbf{r}(t)\), each component (\(x\), \(y\), and \(z\)) is given as a function of \(t\).
• This approach allows the curve to be described in terms of a single parameter, which can simplify calculations and visualizations.
• By using parametric equations, we have the freedom to consider multiple values for \(z\) simultaneously, thereby encompassing variations within the solution.

Understanding parametric equations is critical for analyzing complex curves, such as those created by intersections, in a more manageable way.