We are tasked with finding the point on the surface of a sphere that you are looking at, given that you are standing at the point $P_0 = (2, 3, 0)$ and looking in the direction of another point.

Key Information:

• The sphere has an equation $x^2 + y^2 + z^2 = \sqrt{13}$.
• The given point is either $(-4, -5, 3)$ (if not visible, the direction changes to $(-4, -5, 2)$).

Let’s break it down.

Equation of Sphere

The equation given for the sphere seems incorrect as it is not a typical sphere equation. Assuming it was meant to be $x^2 + y^2 + z^2 = 13$ (standard form of a sphere, where $13$ is the square of the radius, which is $\sqrt{13}$).

Parametric Line Equation

The direction of the line you are looking in, from $P_0$, can be described parametrically as:

$\mathbf{r}(t) = (2, 3, 0) + t((-4 - 2), (-5 - 3), (3 - 0))$

This simplifies to:

$\mathbf{r}(t) = (2, 3, 0) + t(-6, -8, 3)$

Thus, the parametric equation for the line is:

$\mathbf{r}(t) = (2 - 6t, 3 - 8t, 3t)$

Intersection with the Sphere

Substitute $x = 2 - 6t$, $y = 3 - 8t$, and $z = 3t$ into the sphere equation $x^2 + y^2 + z^2 = 13$:

$(2 - 6t)^2 + (3 - 8t)^2 + (3t)^2 = 13$

Now, expand and simplify:

$(4 - 24t + 36t^2) + (9 - 48t + 64t^2) + 9t^2 = 13$ $4 + 9 + 36t^2 + 64t^2 + 9t^2 - 24t - 48t = 13$ $13 + 109t^2 - 72t = 13$ $109t^2 - 72t = 0$ $t(109t - 72) = 0$

So, $t = 0$ or $t = \frac{72}{109}$.

• $t = 0$ gives the point $(2, 3, 0)$, which is the starting point.
• $t = \frac{72}{109}$ gives the point on the sphere.

Coordinates of the Intersection Point

Substitute $t = \frac{72}{109}$ into the parametric line equations:

$x = 2 - 6 \times \frac{72}{109} = 2 - \frac{432}{109} = \frac{218 - 432}{109} = \frac{-214}{109} \approx -1.96330$ $y = 3 - 8 \times \frac{72}{109} = 3 - \frac{576}{109} = \frac{327 - 576}{109} = \frac{-249}{109} \approx -2.28440$ $z = 3 \times \frac{72}{109} = \frac{216}{109} \approx 1.98165$

Final Answer

The point on the sphere where you are looking is approximately $(-1.96330, -2.28440, 1.98165)$.

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Would you like further details or have any questions? Here are 5 follow-up questions related to this problem:

1. How is the parametric equation of a line derived in general?
2. What happens when you plug different values of $t$ into the parametric equation?
3. What are some applications of finding intersection points with spheres?
4. How would the process change if the sphere had a different radius?
5. Can this method be used for other conic sections, like ellipses?

Tip: Always verify the form of a given equation before starting, as errors can propagate through calculations.