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 value(s) of \(k\) is/are:

• A. \(5 \)
• B. \(-5 \)
• C. \(3 \)
• D. \(-3 \)
• E. \(\pm 7 \)

Hint:

Always remember identity: $|\vec{a}\times\vec{b}|^2 + (\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2$.

Answer 1

Answer: (E)

Concept: Identity: \[ |\vec{a}\times\vec{b}|^2 = |\vec{a}|^2|\vec{b}|^2 - (\vec{a}\cdot\vec{b})^2 \]

Step 1: Compare given equation.
\[ |\vec{a}|^2|\vec{b}|^2 - (\vec{a}\cdot\vec{b})^2 = k^2 - (\vec{a}\cdot\vec{b})^2 \]

Step 2: Simplify.
\[ |\vec{a}|^2|\vec{b}|^2 = k^2 \] \[ (7)^2 (1)^2 = k^2 = 49 \] \[ k = \pm 7 \]