To check if the points $A(-4, -1, 6)$, $B(3, 5, 0)$, and $C(-5, -5, -5)$ lie on a straight line, we need to verify if they are collinear. One method is to calculate the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ and see if they are scalar multiples of each other, which would indicate that the points are collinear.

Step 1: Calculate the vector $\overrightarrow{AB}$

$\overrightarrow{AB} = B - A = (3 - (-4), 5 - (-1), 0 - 6) = (3 + 4, 5 + 1, 0 - 6) = (7, 6, -6)$

Step 2: Calculate the vector $\overrightarrow{AC}$

$\overrightarrow{AC} = C - A = (-5 - (-4), -5 - (-1), -5 - 6) = (-5 + 4, -5 + 1, -5 - 6) = (-1, -4, -11)$

Step 3: Check if $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are scalar multiples

We need to check if there exists a scalar $k$ such that: $(7, 6, -6) = k(-1, -4, -11)$

Comparing the components:

1. $7 = k(-1) \implies k = -7$
2. $6 = k(-4) \implies k = -\frac{6}{4} = -\frac{3}{2}$
3. $-6 = k(-11) \implies k = \frac{6}{11}$

Since $k$ is not consistent across all three components, the vectors are not scalar multiples, meaning that the points $A$, $B$, and $C$ are not collinear and do not lie on a straight line.

Let me know if you'd like more details or have questions!

Here are five related questions:

1. How can you find the distance between two points in 3D space?
2. What is the equation of a line in 3D given two points?
3. How do you determine if vectors are parallel?
4. Can you find the area of a triangle formed by three points in 3D space?
5. How do you calculate the cross product of two vectors?

Tip: In 3D geometry, using the cross product of vectors is a great way to check if points are collinear, as collinear vectors will have a zero cross product.