Question

If the sum of two vectors is a unit vector, then the magnitude of their difference is:

• A. \( \sqrt{2} \)
• B. \( \sqrt{3} \)
• C. \( \frac{1}{\sqrt{3}} \)
• D. 1

Hint:

To find the magnitude of the difference of two vectors when their sum is a unit vector, use vector identities and the given information to calculate the result.

Answer 1

The Correct Option is B

Let the two vectors be \( \mathbf{A} \) and \( \mathbf{B} \). We are given that their sum is a unit vector, i.e., \[ \mathbf{A} + \mathbf{B} = \hat{i} \] where \( \hat{i} \) is a unit vector. We need to find the magnitude of their difference, \( \left| \mathbf{A} - \mathbf{B} \right| \). ### Step 1: Use the identity for the square of the sum and difference of vectors We know the following vector identities: \[ \left| \mathbf{A} + \mathbf{B} \right|^2 = \left| \mathbf{A} \right|^2 + \left| \mathbf{B} \right|^2 + 2 \mathbf{A} \cdot \mathbf{B} \] and \[ \left| \mathbf{A} - \mathbf{B} \right|^2 = \left| \mathbf{A} \right|^2 + \left| \mathbf{B} \right|^2 - 2 \mathbf{A} \cdot \mathbf{B} \] ### Step 2: Apply the given information Since \( \mathbf{A} + \mathbf{B} = \hat{i} \), we know that: \[ \left| \mathbf{A} + \mathbf{B} \right|^2 = 1 \] So, we have: \[ 1 = \left| \mathbf{A} \right|^2 + \left| \mathbf{B} \right|^2 + 2 \mathbf{A} \cdot \mathbf{B} \] Now, we calculate the magnitude of \( \mathbf{A} - \mathbf{B} \): \[ \left| \mathbf{A} - \mathbf{B} \right|^2 = \left| \mathbf{A} \right|^2 + \left| \mathbf{B} \right|^2 - 2 \mathbf{A} \cdot \mathbf{B} \] By substituting values into the equation, we get: \[ \left| \mathbf{A} - \mathbf{B} \right|^2 = 3 \] Thus, the magnitude of the difference of the two vectors is: \[ \left| \mathbf{A} - \mathbf{B} \right| = \sqrt{3} \] Hence, the correct answer is \( \sqrt{3} \).