Question

If $a , \,b ,\, c$ are vectors such that $a + b + c =0$ and $| a |=7,| b |=5,| c |=3$, then the angle between $c$ and $b$ is

• A. $\pi / 3 $
• B. $\pi /6$
• C. $\pi / 4$
• D. $\pi $
• E. 0

Answer 1

Answer: (A)

We have,
$a + b + c =0$
$\Rightarrow b + c =- a$
$\Rightarrow | b + c |=|- a |$
$\Rightarrow | b + c |=| a |$
$\Rightarrow | b + c |^{2}=| a |^{2}$
$\Rightarrow ( b + c ) \cdot( b + c )=| a |^{2}$
$\Rightarrow | b |^{2}+| c |^{2}+2| b || c | \cos \,\theta=| a |^{2}$
$\Rightarrow (5)^{2}+(3)^{2}+2 \times 5 \times 3 \cos \,\theta=(7)^{2}$
$\Rightarrow 25+9+30 \cos \,\theta=49$
$\Rightarrow 30 \cos \theta \,=15$
$\Rightarrow \cos \,\theta=\frac{1}{2}$
$\Rightarrow \theta=60^{\circ}$ or $\pi / 3$
$\therefore $ Angle between $b$ and $c$ is $\frac{\pi}{3}$.