If the angle between the lnes with direction ratios (3,4,x) and (2,-1,4) is `pi/2` then the value of x is

Options

A. -2
B. 2
C. `-1/2`
D. `1/2`

Explanation

To find the value of \( x \) such that the angle between the lines with direction ratios \( (3, 4, x) \) and \( (2, -1, 4) \) is \( \frac{\pi}{2} \) (90 degrees), we can follow these steps:

### Step 1: Understand the condition for perpendicular lines
The lines are perpendicular if the angle between them is \( \frac{\pi}{2} \). This means that the dot product of their direction vectors must equal zero.

### Step 2: Define the direction vectors
Let the direction vector of the first line be:
\[ \mathbf{b_1} = 3\mathbf{i} + 4\mathbf{j} + x\mathbf{k} \]
And the direction vector of the second line be:
\[ \mathbf{b_2} = 2\mathbf{i} - 1\mathbf{j} + 4\mathbf{k} \]

### Step 3: Calculate the dot product
The dot product \( \mathbf{b_1} \cdot \mathbf{b_2} \) is given by:
\[ \mathbf{b_1} \cdot \mathbf{b_2} = (3)(2) + (4)(-1) + (x)(4) \]

### Step 4: Set the dot product equal to zero
Since the lines are perpendicular, we set the dot product to zero:
\[ 3 \cdot 2 + 4 \cdot (-1) + x \cdot 4 = 0 \]
This simplifies to:
\[ 6 - 4 + 4x = 0 \]

### Step 5: Simplify the equation
Combine like terms:
\[ 2 + 4x = 0 \]

### Step 6: Solve for \( x \)
Now, isolate \( x \):
\[ 4x = -2 \\ x = -\frac{2}{4} \\ x = -\frac{1}{2} \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{-\frac{1}{2}} \]

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