If the line through the points `A(k,1,-1) and B (2k,0,2)` is perpendicular to the line through the points B and `C(2+2k,k,1)`, then what is the value of k?

Options

A. `-1`
B. 1
C. `-3`
D. 3

Answer

To find the value of \( k \) such that the line through points \( A(k, 1, -1) \) and \( B(2k, 0, 2) \) is perpendicular to the line through points \( B \) and \( C(2 + 2k, k, 1) \), we will follow these steps:

### Step 1: Find the vector \( \overrightarrow{AB} \)

The vector \( \overrightarrow{AB} \) can be calculated as follows:
\[ \overrightarrow{AB} = B - A = (2k - k, 0 - 1, 2 - (-1)) = (k, -1, 3) \]

### Step 2: Find the vector \( \overrightarrow{BC} \)

Next, we calculate the vector \( \overrightarrow{BC} \):
\[ \overrightarrow{BC} = C - B = ((2 + 2k) - 2k, k - 0, 1 - 2) = (2, k, -1) \]

### Step 3: Use the condition for perpendicularity

The vectors \( \overrightarrow{AB} \) and \( \overrightarrow{BC} \) are perpendicular if their dot product is zero:
\[ \overrightarrow{AB} \cdot \overrightarrow{BC} = 0 \]
Calculating the dot product:
\[ (k, -1, 3) \cdot (2, k, -1) = k \cdot 2 + (-1) \cdot k + 3 \cdot (-1) \]
This simplifies to:
\[ 2k - k - 3 = 0 \]
or
\[ k - 3 = 0 \]

### Step 4: Solve for \( k \)

From the equation \( k - 3 = 0 \), we find:
\[ k = 3 \]

### Conclusion

The value of \( k \) is \( 3 \).

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