Using vectors, find the value of k, such that the points (k,-10, 3), (1,-1, 3) and (3, 5, 3) are collinear.

Answer

To find the value of \( k \) such that the points \( A(k, -10, 3) \), \( B(1, -1, 3) \), and \( C(3, 5, 3) \) are collinear, we can use vector analysis. Here’s the step-by-step solution:

### Step 1: Define the Points
Let the points be defined as:
- \( A(k, -10, 3) \)
- \( B(1, -1, 3) \)
- \( C(3, 5, 3) \)

### Step 2: Find the Vectors \( \overrightarrow{AB} \) and \( \overrightarrow{BC} \)
To find the vectors, we can use the formula:
\[ \overrightarrow{AB} = \overrightarrow{B} - \overrightarrow{A} \]
\[ \overrightarrow{BC} = \overrightarrow{C} - \overrightarrow{B} \]

#### Calculate \( \overrightarrow{AB} \):
\[ \overrightarrow{AB} = (1 - k, -1 - (-10), 3 - 3) = (1 - k, 9, 0) \]

#### Calculate \( \overrightarrow{BC} \):
\[ \overrightarrow{BC} = (3 - 1, 5 - (-1), 3 - 3) = (2, 6, 0) \]

### Step 3: Set Up the Collinearity Condition
For the points to be collinear, the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{BC} \) must be scalar multiples of each other. This means:
\[ \overrightarrow{AB} = \lambda \overrightarrow{BC} \]
for some scalar \( \lambda \).

### Step 4: Set Up the Equations
From the vectors, we have:
\[ (1 - k, 9, 0) = \lambda (2, 6, 0) \]

This gives us the following equations:
1. \( 1 - k = 2\lambda \) (for the \( i \) component)
2. \( 9 = 6\lambda \) (for the \( j \) component)

### Step 5: Solve for \( \lambda \)
From the second equation:
\[ 9 = 6\lambda \implies \lambda = \frac{9}{6} = \frac{3}{2} \]

### Step 6: Substitute \( \lambda \) Back to Find \( k \)
Substituting \( \lambda \) back into the first equation:
\[ 1 - k = 2\left(\frac{3}{2}\right) \]
\[ 1 - k = 3 \]
\[ -k = 3 - 1 \]
\[ -k = 2 \implies k = -2 \]

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{-2} \]