The following three points lie on a straight line:
A(-4,-1,6), B(3,5,0), C(-5,-5,-5)

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.

The following three points lie on a straight line: A(-4,-1,6), B(3,5,0), C(-5,-5,-5)

Learn how to check if three points in 3D space are collinear using vector subtraction and scalar multiples. This problem involves points A(-4,-1,6), B(3,5,0), and C(-5,-5,-5) and provides a step-by-step explanation of how to verify collinearity in 3D geometry. Suitable for high school students (Grades 10-12).

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