Question

If $\vec{a} + \vec{b} + \vec{c} = 0$, and $|\vec{a}| = 7$, $|\vec{b}| = 5$, $|\vec{c}| = 3$, then the angle between $\vec{b}$ and $\vec{c}$ is

• A. $30^\circ$
• B. $45^\circ$
• C. $60^\circ$
• D. $90^\circ$

Hint:

Use vector identities and dot product expansion for angle calculations involving sums.

Answer 1

The Correct Option is C

Given $\vec{a} + \vec{b} + \vec{c} = 0 \Rightarrow \vec{a} = -(\vec{b} + \vec{c})$
Now take magnitude squared:
$|\vec{a}|^2 = |\vec{b} + \vec{c}|^2 = \vec{b} \cdot \vec{b} + \vec{c} \cdot \vec{c} + 2\vec{b} \cdot \vec{c}$
$\Rightarrow 49 = 25 + 9 + 2 \cdot |\vec{b}||\vec{c}| \cos\theta$
$49 = 34 + 30 \cos\theta \Rightarrow 15 = 30 \cos\theta \Rightarrow \cos\theta = \dfrac{1}{2}$
$\Rightarrow \theta = 60^\circ$