Question

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter. $$\text { Surfaces } \quad \text { Parameter }$$ $$ x^{2}+y^{2}+z^{2}=16, x y=4 \quad x=t \text { (first octant) } $$

Short answer

The space curve, represented by the intersection of the surfaces \(x^{2}+y^{2}+z^{2}=16\) and \(xy=4\), can be represented by the vector-valued function \(\mathbf{r}(t)=\begin{bmatrix} t \ 4/t \ \sqrt{16-t^2} \end{bmatrix}\) where \(x = t\), \(y = 4/t\), and \(z = \sqrt{16 - t^2}\).

Step-by-step solution

Intersection of the surfaces

Solve the equations \(x^{2}+y^{2}+z^{2}=16\) and \(xy=4\), simultaneously. From the second equation, we can express \(y\) as a function of \(x\), that is \(y=4/x\). Now, substitute \(y\) into the first equation to get \(x^{2}+(4/x)^{2}+z^{2}=16\). This can be simplified to \(x^{2}(1+(4/x)^{2})+z^{2}=16\), which in turn simplifies to \(x^{2}+16/x^2+z^{2}=16\). This can be further simplified to \(x^{4}+16+z^{2}x^{2}=16x^{2}\). After further simplification, we get \(x^{4}+z^{2}x^{2}-16x^{2}=0\). We can factor the resulting equation to get \(x^{2}(x^{2}+z^{2}-16)=0\). Hence, \(x^{2}=0\) or \(x^{2}+z^{2}-16=0\). Since we're in the first octant, \(x\neq0\), so we have \(x^{2}+z^{2}=16\)

Parametrize the curve

Now, we substitute \(x=t\) into the equations obtained from step 1. We get \(y=4/t\) from the equation \(y=4/x\) and from \(x^{2}+z^{2}=16\) we get \(z=\sqrt{16-t^2}\) since \(z\) must be positive in the first octant.

Vector-valued function

The vector-valued function that represents the curve is then formed by combining the parameterized equations. This function will be in the form \(\mathbf{r}(t)=\begin{bmatrix} x(t) \ y(t) \ z(t) \end{bmatrix}= \begin{bmatrix} t \ 4/t \ \sqrt{16-t^2} \end{bmatrix}\). This represents the intersection of the two given surfaces in the first octant.

Key concepts

Space Curve

Imagine you’re running along a twisty, bumpy path through a hillside park. Your journey along this path can be thought of as a space curve – a type of curve that snakes through three-dimensional space. In mathematics, a space curve helps us visualize the complex movement and location of an object as it moves through space, not just along a flat surface.

In the exercise we’re considering, the space curve occurs at the intersection of two surfaces, namely a sphere (given by the equation \(x^{2}+y^{2}+z^{2}=16\) ) and a hyperbolic cylinder (arising from the equation \(xy=4\)). Sketching a space curve can be quite intricate, as it requires you to picture a three-dimensional form on paper, which is naturally two-dimensional. Students often find it easier to grapple with these concepts when they use physical models or computer graphics to visualize the curves in three dimensions.

To further clarify, let’s tie this concept back to the exercise. The space curve formed by the intersection of the sphere and the hyperbolic cylinder is not just any random curve. It's defined strictly by points that satisfy both surface equations at the same time, and it is contained within the shared space where these two surfaces overlap.

Parametric Equations

Switching gears from space curves, let's talk about how to describe them. One effective way to characterize a space curve is through parametric equations. Think of these as a way to give you 'directions' on how to trace out the path of the curve. Instead of saying, 'Meet me at the park near the big oak tree and the red bench,' parametric equations give exact instructions like, 'Start here, walk north 20 steps, then east 10 steps'. In mathematics, these are precise, coordinate-based directions for plotting the curve.

In the context of our exercise, we use the parameter \(x=t\) to describe how the curve moves through space. By choosing \(t\) as a parameter, each variable in our system – \(x, y, z\) – becomes a function of \(t\). Thus, we have \(x(t) = t\), \(y(t)=4/t\), and for \(z(t)\), we use the positive root of the remaining equation to get \(z(t) = \(16-t^2\)\) because we're describing the curve in the first octant where all coordinates are positive.

Crafting parametric equations requires a thoughtful approach, simplifying complex surface equations to be expressed in terms of a single variable (in this case, \(t\)) This paves the way for the final step – creating a vector-valued function that encapsulates the complete description of our space curve.

Surfaces Intersection

Moving to an intersection more complex than any street crossing, the intersection of surfaces in a three-dimensional space is where two or more surfaces coincide. This intersection can take many shapes: it might be a line, a curve, or potentially a more complex figure. It depends entirely on the nature of the surfaces involved. Just as if you had two semi-transparent sheets with shapes drawn on them, the intersection would be where the shapes overlap.

In our exercise, the surfaces – a sphere and a hyperbolic cylinder – cross paths in a special way. By solving their equations together, as we did in the step-by-step solution, we find the space curve that represents their intersection. In the real world, understanding surfaces intersection can be crucial in fields ranging from architecture to engineering, where it’s often necessary to know exactly where different parts of a structure or machine line up. Clear comprehension of this concept helps students better solve the problem at hand, making the mathematical journey from the abstract to the concrete much smoother.