Given : `vecA=hati+hatjandvecB=hati+hatk`. What is the value of the scalar product of A and B?

Options

A. 1
B. `sqrt(2)`
C. `sqrt(3)`
D. 2

Answer

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

1. **Identify the vectors**:
\[ \vec{A} = \hat{i} + \hat{j} \]
\[ \vec{B} = \hat{i} + \hat{k} \]

2. **Write the formula for the scalar product**:
The scalar product (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 \(A_x\), \(A_y\), and \(A_z\) are the components of vector \(\vec{A}\), and \(B_x\), \(B_y\), and \(B_z\) are the components of vector \(\vec{B}\).

3. **Extract the components**:
From \(\vec{A} = \hat{i} + \hat{j}\):
- \(A_x = 1\) (coefficient of \(\hat{i}\))
- \(A_y = 1\) (coefficient of \(\hat{j}\))
- \(A_z = 0\) (no \(\hat{k}\) component)

From \(\vec{B} = \hat{i} + \hat{k}\):
- \(B_x = 1\) (coefficient of \(\hat{i}\))
- \(B_y = 0\) (no \(\hat{j}\) component)
- \(B_z = 1\) (coefficient of \(\hat{k}\))

4. **Substitute the components into the formula**:
\[ \vec{A} \cdot \vec{B} = (1)(1) + (1)(0) + (0)(1) \]

5. **Calculate the result**:
\[ \vec{A} \cdot \vec{B} = 1 + 0 + 0 = 1 \]

Thus, the value of the scalar product of \(\vec{A}\) and \(\vec{B}\) is:
\[ \boxed{1} \]