Let's solve the problem step by step.

Step 1: Determine if the point $(-2, -3, 2)$ is visible from $(5, 6, 0)$

• The opaque ball is centered at the origin $(0, 0, 0)$ with a radius of 1.
• To determine if $(-2, -3, 2)$ is visible from $(5, 6, 0)$, we need to check if the line segment connecting these two points intersects the sphere.

Equation of the line:

The line passing through $(5, 6, 0)$ and $(-2, -3, 2)$ can be parameterized as: $\mathbf{r}(t) = (1-t) \mathbf{r_1} + t \mathbf{r_2}$ where $\mathbf{r_1} = (5, 6, 0)$ and $\mathbf{r_2} = (-2, -3, 2)$, and $t$ is a parameter.

So the equation of the line is: $\mathbf{r}(t) = (5 + t(-7), 6 + t(-9), 0 + t(2))$ which simplifies to: $\mathbf{r}(t) = (5 - 7t, 6 - 9t, 2t)$

Step 2: Check for intersection with the sphere

The equation of the sphere is: $x^2 + y^2 + z^2 = 1$

Substitute the parametric coordinates of the line into the sphere's equation: $(5 - 7t)^2 + (6 - 9t)^2 + (2t)^2 = 1$ Expanding and solving this quadratic equation will tell us whether the line intersects the sphere.

Step 3: Solve the quadratic equation

Expand and simplify: $(5 - 7t)^2 = 25 - 70t + 49t^2$ $(6 - 9t)^2 = 36 - 108t + 81t^2$ $(2t)^2 = 4t^2$ Adding these up: $(25 - 70t + 49t^2) + (36 - 108t + 81t^2) + 4t^2 = 1$ $25 + 36 - 1 - 70t - 108t + (49t^2 + 81t^2 + 4t^2) = 0$ $60 - 178t + 134t^2 = 0$

This is a quadratic equation in $t$: $134t^2 - 178t + 60 = 0$

Solve for $t$ using the quadratic formula: $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 134$, $b = -178$, and $c = 60$.

Step 4: Determine the point on the sphere

If the discriminant $b^2 - 4ac$ is positive, the line intersects the sphere, and we can calculate the intersection points. If it's negative, there's no intersection, and the point $(-2, -3, 2)$ is visible from $(5, 6, 0)$.

Let me calculate this quadratic equation and find the solution for you.The discriminant of the quadratic equation is $-476$, which is negative. This means that the line segment connecting the points $(5, 6, 0)$ and $(-2, -3, 2)$ does not intersect the sphere.

Conclusion:

• The point $(-2, -3, 2)$ is visible from $(5, 6, 0)$.
• Since the point $(-2, -3, 2)$ is visible, the direction $(-2, -3, 2)$ does not intersect the sphere, and the intersection calculation isn't needed for finding another point.

Would you like more details or have any questions?

Related Questions:

1. How do you find the intersection points between a line and a sphere?
2. What is the significance of the discriminant in quadratic equations?
3. How can you determine the visibility of a point relative to an object?
4. What are the steps to parameterize a line in 3D space?
5. How do you compute the direction vector between two points?

Tip:

When dealing with visibility problems involving spherical objects, always start by checking the discriminant of the intersection equation to determine whether a line actually intersects the sphere.