Question

If \( |\vec{a}| = 10, |\vec{b}| = 2 \) and \( \vec{a} \cdot \vec{b} = 12 \), then the value of \( |\vec{a} \times \vec{b}| \) is:

• A. 10
• B. 14
• C. 16
• D. 5

Hint:

The magnitude of the cross product \( |\vec{a} \times \vec{b}| \) can be found using the formula: \[ |\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2 \]

Answer 1

The Correct Option is C

We are given that: \[ |\vec{a}| = 10, \quad |\vec{b}| = 2, \quad \vec{a} \cdot \vec{b} = 12 \] We use the identity: \[ |\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2 \] Substituting the given values: \[ |\vec{a} \times \vec{b}|^2 = (10^2)(2^2) - (12^2) \] \[ = (100)(4) - 144 = 400 - 144 = 256 \] Thus: \[ |\vec{a} \times \vec{b}| = \sqrt{256} = 16 \] Therefore, the value of \( |\vec{a} \times \vec{b}| \) is \( 16 \).