Identify an example where A · B = C · B while A ≠ C .

Identify an example where

\vec{A} \cdot \vec{B} = \vec{C} \cdot \vec{B}

while

\vec{A} \neq \vec{C}

Options

A $\vec{A} = \hat{i} + \hat{j}$ , $\vec{B} = \hat{i}$ , $\vec{C} = \hat{i} - \hat{j}$
B $\vec{A} = 2\hat{i} + \hat{j}$ , $\vec{B} = \hat{i}$ , $\vec{C} = \hat{i} - \hat{j}$
C $\vec{A} = \hat{i} + \hat{j}$ , $\vec{B} = \hat{i}$ , $\vec{C} = \hat{i} + \hat{j}$
D $\vec{A} = \hat{i}$ , $\vec{B} = \hat{i}$ , $\vec{C} = \hat{k}$

A. $\vec{A} = \hat{i} + \hat{j}$ , $\vec{B} = \hat{i}$ , $\vec{C} = \hat{i} - \hat{j}$

Checking the given options for the condition

\vec{A} \cdot \vec{B} = \vec{C} \cdot \vec{B}

and

\vec{A} \neq \vec{C}

.

For the first option:

\vec{A} \cdot \vec{B} = (\hat{i} + \hat{j}) \cdot \hat{i} = 1

\vec{C} \cdot \vec{B} = (\hat{i} - \hat{j}) \cdot \hat{i} = 1

Here,

\vec{A} \cdot \vec{B} = \vec{C} \cdot \vec{B} = 1

.
Also,

\hat{i} + \hat{j} \neq \hat{i} - \hat{j}

, so

\vec{A} \neq \vec{C}

.

For the second option:

\vec{A} \cdot \vec{B} = (2\hat{i} + \hat{j}) \cdot \hat{i} = 2

\vec{C} \cdot \vec{B} = (\hat{i} - \hat{j}) \cdot \hat{i} = 1

Here,

\vec{A} \cdot \vec{B} \neq \vec{C} \cdot \vec{B}

.

For the third option:

\vec{A} = \hat{i} + \hat{j}

and

\vec{C} = \hat{i} + \hat{j}

Here,

\vec{A} = \vec{C}

, which violates the given condition.

For the fourth option:

\vec{A} \cdot \vec{B} = \hat{i} \cdot \hat{i} = 1

\vec{C} \cdot \vec{B} = \hat{k} \cdot \hat{i} = 0

Here,

\vec{A} \cdot \vec{B} \neq \vec{C} \cdot \vec{B}

.

The first option satisfies all the given conditions.