Which of the following is an example of two different vectors with same magnitude ?

Options

A. `(2hati + 3hatj + hatk) " and " (2hati + 3hatj - hatk)`
B. `(3hati + 5hatj + hatk) " and " (3hati + 4hatj + hatk)`
C. `(hatj + hatk) " and " (2hatj + 3hatk)`
D. none of these

Answer

To solve the question of identifying two different vectors with the same magnitude, we will analyze the options provided. Let's consider option A as described in the video transcript.

### Step-by-Step Solution:

1. **Identify the Vectors**:
- Let the first vector be \( \mathbf{A} = 2\hat{i} + 3\hat{j} + 1\hat{k} \).
- Let the second vector be \( \mathbf{B} = 2\hat{i} + 3\hat{j} - 1\hat{k} \).

2. **Calculate the Magnitude of Vector A**:
- The magnitude of vector \( \mathbf{A} \) is calculated using the formula:
\[ |\mathbf{A}| = \sqrt{(2^2) + (3^2) + (1^2)} \]
- Calculating it:
\[ |\mathbf{A}| = \sqrt{4 + 9 + 1} = \sqrt{14} \]

3. **Calculate the Magnitude of Vector B**:
- The magnitude of vector \( \mathbf{B} \) is calculated similarly:
\[ |\mathbf{B}| = \sqrt{(2^2) + (3^2) + (-1^2)} \]
- Calculating it:
\[ |\mathbf{B}| = \sqrt{4 + 9 + 1} = \sqrt{14} \]

4. **Compare the Magnitudes**:
- From the calculations, we find:
\[ |\mathbf{A}| = \sqrt{14} \quad \text{and} \quad |\mathbf{B}| = \sqrt{14} \]
- Both vectors have the same magnitude.

5. **Check if the Vectors are Different**:
- The vector \( \mathbf{A} \) has a positive \( \hat{k} \) component, while vector \( \mathbf{B} \) has a negative \( \hat{k} \) component.
- Therefore, \( \mathbf{A} \neq \mathbf{B} \).

### Conclusion:
The vectors \( \mathbf{A} \) and \( \mathbf{B} \) are different vectors that have the same magnitude. Thus, option A represents two different vectors with the same magnitude.