If `vec(A)=-hat(i)+ x hat(j) - 3 hat(k)` and `vec(B) = 2hat(i) - hat(k)` then the value(s) of `x` for which the dot product of `bar(A)` & `bar(B)` is equal to unity :

Options

A. `-1`
B. `1`
C. `2`
D. `-3`

Explanation

To solve the problem, we need to find the values of \( x \) for which the dot product of the vectors \( \vec{A} \) and \( \vec{B} \) is equal to 1.

Given:
\[ \vec{A} = -\hat{i} + x\hat{j} - 3\hat{k} \]
\[ \vec{B} = 2\hat{i} - \hat{k} \]

### Step 1: Write the dot product formula
The dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is given by:
\[ \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z \]
where \( A_x, A_y, A_z \) are the components of \( \vec{A} \) and \( B_x, B_y, B_z \) are the components of \( \vec{B} \).

### Step 2: Identify the components
From the vectors:
- For \( \vec{A} \):
- \( A_x = -1 \)
- \( A_y = x \)
- \( A_z = -3 \)

- For \( \vec{B} \):
- \( B_x = 2 \)
- \( B_y = 0 \) (since there is no \( \hat{j} \) component)
- \( B_z = -1 \)

### Step 3: Calculate the dot product
Now, substituting the components into the dot product formula:
\[ \vec{A} \cdot \vec{B} = (-1)(2) + (x)(0) + (-3)(-1) \]
Calculating this gives:
\[ \vec{A} \cdot \vec{B} = -2 + 0 + 3 = 1 \]

### Step 4: Set the dot product equal to unity
We need the dot product to equal 1:
\[ 1 = 1 \]

### Step 5: Analyze the result
The equation \( 1 = 1 \) is always true, which means that the value of \( x \) does not affect the result of the dot product. Therefore, any value of \( x \) will satisfy the condition.

### Conclusion
The values of \( x \) for which the dot product of \( \vec{A} \) and \( \vec{B} \) is equal to unity are all real numbers.