If the vectors `veca = 2hati -4hatj-2hatk` and `vecb = 3hati +2hatj+xhatk` are at the right angles to each other, then the value of r should be

Options

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

Answer

To find the value of \( x \) such that the vectors \( \vec{a} = 2\hat{i} - 4\hat{j} - 2\hat{k} \) and \( \vec{b} = 3\hat{i} + 2\hat{j} + x\hat{k} \) are at right angles to each other, we can use the property that the dot product of two perpendicular vectors is zero.

### Step-by-Step Solution:

1. **Write the Dot Product Formula**:
The dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:
\[ \vec{a} \cdot \vec{b} = (a_x \cdot b_x) + (a_y \cdot b_y) + (a_z \cdot b_z) \]
where \( a_x, a_y, a_z \) are the components of \( \vec{a} \) and \( b_x, b_y, b_z \) are the components of \( \vec{b} \).

2. **Substitute the Components**:
For \( \vec{a} = 2\hat{i} - 4\hat{j} - 2\hat{k} \) and \( \vec{b} = 3\hat{i} + 2\hat{j} + x\hat{k} \), the components are:
- \( a_x = 2, a_y = -4, a_z = -2 \)
- \( b_x = 3, b_y = 2, b_z = x \)

3. **Calculate the Dot Product**:
Substitute the components into the dot product formula:
\[ \vec{a} \cdot \vec{b} = (2 \cdot 3) + (-4 \cdot 2) + (-2 \cdot x) \]
Simplifying this gives:
\[ \vec{a} \cdot \vec{b} = 6 - 8 - 2x \]
\[ \vec{a} \cdot \vec{b} = -2 - 2x \]

4. **Set the Dot Product to Zero**:
Since the vectors are perpendicular, we set the dot product equal to zero:
\[ -2 - 2x = 0 \]

5. **Solve for \( x \)**:
Rearranging the equation gives:
\[ -2x = 2 \]
Dividing both sides by -2 results in:
\[ x = -1 \]

### Final Answer:
The value of \( x \) should be \( -1 \).