Question

If \( \theta \) is the angle between two vectors \( \vec{a} \) and \( \vec{b} \) such that \( |\vec{a}| = 7 \), \( |\vec{b}| = 1 \) and \( |\vec{a} \times \vec{b}|^2 = k^2 - (\vec{a} \cdot \vec{b})^2 \), then the values of \( k \) and \( \theta \) are:

• A. \( k=1, \theta=45^\circ \)
• B. \( k=7, \theta=60^\circ \)
• C. \( k=49, \theta=90^\circ \)
• D. \( k=7 \) and \( \theta \) is arbitrary

Hint:

Whenever an angle \( \theta \) disappears from the final simplified equation, it means \( \theta \) is arbitrary and not uniquely determined by the given conditions.

Answer 1

The Correct Option is D

Step 1: Given Information.
\n\nWe are given:
\n \( |\vec{a}| = 7 \)
\n \( |\vec{b}| = 1 \)
\n \( |\vec{a} \times \vec{b}|^2 = k^2 - (\vec{a} \cdot \vec{b})^2 \)
\n\n
Step 2: Express knowns.
\n\nWe know:\n\[\n|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta = 7\sin\theta\n\]\nthus:\n\[\n|\vec{a} \times \vec{b}|^2 = 49\sin^2\theta\n\]\n\nAlso:\n\[\n\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = 7\cos\theta\n\quad \Rightarrow \quad\n(\vec{a} \cdot \vec{b})^2 = 49\cos^2\theta\n\]\n\nTherefore, the given condition becomes:\n\[\n49\sin^2\theta = k^2 - 49\cos^2\theta\n\]\n\n
Step 3: Simplify.
\n\nUsing the identity:\n\[\n\sin^2\theta + \cos^2\theta = 1\n\quad \Rightarrow \quad\n\sin^2\theta = 1 - \cos^2\theta\n\]\n\nThus:\n\[\n49(1 - \cos^2\theta) = k^2 - 49\cos^2\theta\n\]\nExpanding:\n\[\n49 - 49\cos^2\theta = k^2 - 49\cos^2\theta\n\]\n\nCancelling \( -49\cos^2\theta \) from both sides:\n\[\n49 = k^2\n\quad \Rightarrow \quad\nk = 7\n\quad (\text{taking positive value})\n\]\n\n
Step 4: Analyze \( \theta \).
\n\nObserve that \( \theta \) is eliminated during the calculation.
\n\n \( \theta \) is not uniquely determined.
\n \( \theta \) can be any value.
\n\nThus, \( \theta \) can be any value, and \( k = 7 \).\n\n
Step 5: Conclusion.
\n\nTherefore, \( k=7 \) and \( \theta \) is arbitrary.