For what value of x , will the two vector `A= 2hati + 2hatj -x hatk and B =2 hati - hatj - 3hatk ` are perpendicular to each other ?

Options

A. `x=-2//3`
B. `x=-3//2`
C. `x=-4//3`
D. `x=-2//3`

Explanation

To determine the value of \( x \) for which the vectors \( \mathbf{A} = 2\hat{i} + 2\hat{j} - x\hat{k} \) and \( \mathbf{B} = 2\hat{i} - \hat{j} - 3\hat{k} \) are perpendicular, we need to use the property that two vectors are perpendicular if their dot product is zero.

### Step-by-step Solution:

1. **Write down the vectors**:
\[ \mathbf{A} = 2\hat{i} + 2\hat{j} - x\hat{k} \]
\[ \mathbf{B} = 2\hat{i} - \hat{j} - 3\hat{k} \]

2. **Calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \)**:
The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is calculated as follows:
\[ \mathbf{A} \cdot \mathbf{B} = (2\hat{i} + 2\hat{j} - x\hat{k}) \cdot (2\hat{i} - \hat{j} - 3\hat{k}) \]
Using the distributive property of the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = 2 \cdot 2 + 2 \cdot (-1) + (-x) \cdot (-3) \]

3. **Simplify the dot product**:
\[ \mathbf{A} \cdot \mathbf{B} = 4 - 2 + 3x \]
\[ \mathbf{A} \cdot \mathbf{B} = 2 + 3x \]

4. **Set the dot product equal to zero**:
For the vectors to be perpendicular, we set the dot product to zero:
\[ 2 + 3x = 0 \]

5. **Solve for \( x \)**:
\[ 3x = -2 \]
\[ x = -\frac{2}{3} \]

### Final Answer:
The value of \( x \) for which the vectors \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular is:
\[ x = -\frac{2}{3} \]