To find a vector function that represents the curve of intersection of the two surfaces, we need to parametrize both surfaces simultaneously. The two surfaces given are:

1. The cylinder: $x^2 + y^2 = 9$
2. The surface: $z = x^2 - y^2$

Step 1: Parametrizing the Cylinder

The equation $x^2 + y^2 = 9$ suggests a circle in the $xy$-plane with radius 3. This can be parametrized using trigonometric functions:

$x = 3 \cos(t), \quad y = 3 \sin(t)$

where $t$ is the parameter, and $t \in [0, 2\pi]$ covers the full circle.

Step 2: Using the Surface Equation for $z$

The surface equation is $z = x^2 - y^2$. Substituting the parametric expressions for $x$ and $y$ from the cylinder into this equation:

$z = (3 \cos(t))^2 - (3 \sin(t))^2 = 9 \cos^2(t) - 9 \sin^2(t)$

Using the trigonometric identity $\cos(2t) = \cos^2(t) - \sin^2(t)$, this simplifies to:

$z = 9 (\cos^2(t) - \sin^2(t)) = 9 \cos(2t)$

Step 3: Forming the Vector Function

Now, we have parametric expressions for $x$, $y$, and $z$:

$x = 3 \cos(t), \quad y = 3 \sin(t), \quad z = 9 \cos(2t)$

Thus, the vector function that represents the curve of intersection is:

$\mathbf{r}(t) = \langle 3 \cos(t), 3 \sin(t), 9 \cos(2t) \rangle$

where $t \in [0, 2\pi]$.

Final Answer:

The vector function is:

$\mathbf{r}(t) = \langle 3 \cos(t), 3 \sin(t), 9 \cos(2t) \rangle$

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Would you like more details on any of the steps, or do you have any other questions?

Here are 5 related questions:

1. How would you parametrize a general cylinder $x^2 + y^2 = r^2$?
2. What is the significance of trigonometric identities in parametrizing surfaces?
3. How do we parametrize a different surface, such as $z = x + y$, intersecting a cylinder?
4. How would the vector function change if the cylinder had an elliptical cross-section?
5. Can the method of parametrization be applied to three surfaces?

Tip: Always look for symmetry in the surfaces to simplify the parametrization process, such as using trigonometric functions for circular shapes.