The scalar product of two vectors `A=2hati +2hatj -hatk and B=-hatj +hatk` ,is given by

Options

A. A.B=3
B. A.B=4
C. A.B=-4
D. A.B=-3

Explanation

To find the scalar product (dot product) of the two vectors \( \mathbf{A} \) and \( \mathbf{B} \), we can follow these steps:

### Step 1: Write down the vectors
The vectors are given as:
\[ \mathbf{A} = 2\hat{i} + 2\hat{j} - \hat{k} \]
\[ \mathbf{B} = -\hat{j} + \hat{k} \]

### Step 2: Use the formula for the dot product
The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z \]
where \( A_x, A_y, A_z \) are the components of vector \( \mathbf{A} \) and \( B_x, B_y, B_z \) are the components of vector \( \mathbf{B} \).

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

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

### Step 4: Substitute the components into the formula
Now, substituting the components into the dot product formula:
\[ \mathbf{A} \cdot \mathbf{B} = (2)(0) + (2)(-1) + (-1)(1) \]

### Step 5: Calculate each term
Calculating each term:
- The first term: \( 2 \cdot 0 = 0 \)
- The second term: \( 2 \cdot -1 = -2 \)
- The third term: \( -1 \cdot 1 = -1 \)

### Step 6: Add the results
Now, adding these results together:
\[ \mathbf{A} \cdot \mathbf{B} = 0 - 2 - 1 = -3 \]

### Conclusion
Thus, the scalar product of the two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is:
\[ \mathbf{A} \cdot \mathbf{B} = -3 \]

### Final Answer
The answer is \(-3\).

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