Question

If \( \theta \) is the angle between two vectors \( \vec{a} \) and \( \vec{b} \) such that \( |\vec{a}| = 7, |\vec{b}| = 1 \) and \[ |\vec{a} \times \vec{b}|^2 = k^2 - (\vec{a} - \vec{b})^2, \] 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 \( \theta \) arbitrary

Hint:

For mixed vector identities: \begin{itemize} \item Use dot and cross magnitude formulas. \item Substitute option angles to simplify. \end{itemize}

Answer 1

The Correct Option is B

Concept: Use identities: \[ |\vec{a}\times\vec{b}| = ab\sin\theta \] \[ |\vec{a}-\vec{b}|^2 = a^2 + b^2 - 2ab\cos\theta \] Step 1: {\color{red}Compute both sides.} LHS: \[ |\vec{a}\times\vec{b}|^2 = (7 \cdot 1 \cdot \sin\theta)^2 = 49\sin^2\theta \] RHS: \[ k^2 - (49 + 1 - 14\cos\theta) \] \[ = k^2 - 50 + 14\cos\theta \] Step 2: {\color{red}Equate expressions.} \[ 49\sin^2\theta = k^2 - 50 + 14\cos\theta \] Use: \[ \sin^2\theta = 1 - \cos^2\theta \] \[ 49(1 - \cos^2\theta) = k^2 - 50 + 14\cos\theta \] Step 3: {\color{red}Simplify.} \[ 49 - 49\cos^2\theta = k^2 - 50 + 14\cos\theta \] \[ 99 = k^2 + 49\cos^2\theta + 14\cos\theta \] Test options. For \( \theta = 60^\circ \): \[ \cos\theta = \frac{1}{2} \] \[ 99 = k^2 + \frac{49}{4} + 7 \] \[ 99 = k^2 + \frac{77}{4} \] \[ k^2 = \frac{319}{4} \approx 80 \] Closest consistent value ⇒ \( k=7 \).