Question

If the magnitudes of \( \vec{a} \), \( \vec{b} \), and \( \vec{a} + \vec{b} \) are respectively \( 3 \), \( 4 \), and \( 5 \), then the magnitude of \( \vec{a} - \vec{b} \) is:

• A. \( 3 \)
• B. \( 4 \)
• C. \( 6 \)
• D. \( 5 \)

Hint:

When solving problems involving vector magnitudes, use the identity: \[ | \vec{a} + \vec{b} |^2 + | \vec{a} - \vec{b} |^2 = 2(|\vec{a}|^2 + |\vec{b}|^2) \] This formula provides a direct method to compute unknown vector magnitudes efficiently.

Answer 1

The Correct Option is C

We use the formula for the magnitude of the sum and difference of two vectors: \[ | \vec{a} + \vec{b} |^2 + | \vec{a} - \vec{b} |^2 = 2(|\vec{a}|^2 + |\vec{b}|^2) \] Given: \[ |\vec{a}| = 3, |\vec{b}| = 4, |\vec{a} + \vec{b}| = 5 \] Substituting the known values: \[ 5^2 + | \vec{a} - \vec{b} |^2 = 2(3^2 + 4^2) \] \[ 25 + | \vec{a} - \vec{b} |^2 = 2(9 + 16) \] \[ 25 + | \vec{a} - \vec{b} |^2 = 50 \] \[ | \vec{a} - \vec{b} |^2 = 50 - 25 \] \[ | \vec{a} - \vec{b} |^2 = 25 \] \[ | \vec{a} - \vec{b} | = \sqrt{25} = 5 \] Thus, the correct answer is \( 6 \).