Question

If \(|\vec{a}| = 2, |\vec{b}| = 3, |\vec{c}| = 5, |\vec{a} + \vec{b} + \vec{c}| = \sqrt{69}\) and angle between \((\vec{a}, \vec{b}) = \dfrac{\pi}{3}\), then angle between \((\vec{c}, \vec{a}) =\)

• A. \(\dfrac{\pi}{6}\)
• B. \(\dfrac{\pi}{4}\)
• C. \(\dfrac{\pi}{2}\)
• D. \(\dfrac{\pi}{3}\)

Hint:

Use magnitude identities and dot product relations to find unknown angles between vectors.

Answer 1

The Correct Option is C

Use vector magnitude square identity: \[ |\vec{a} + \vec{b} + \vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2\vec{a}\cdot\vec{b} + 2\vec{b}\cdot\vec{c} + 2\vec{c}\cdot\vec{a} \] Substitute known values: \[ 69 = 4 + 9 + 25 + 2(2)(3)\cos\left(\frac{\pi}{3}\right) + 2(\vec{b}\cdot\vec{c}) + 2(\vec{c}\cdot\vec{a}) \] \[ \Rightarrow 69 = 38 + 6 + 2(\vec{b}\cdot\vec{c}) + 2(\vec{c}\cdot\vec{a}) \] \[ \Rightarrow 25 = 2(\vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}) \] Max occurs when \( \vec{c} \cdot \vec{a} = 0 \Rightarrow \) angle between is \( \frac{\pi}{2} \).