Question

$\vec{a},\vec{b},\vec{c}$ are three vectors, such that $\vec{a}+\vec{b}+\vec{c}=\vec{0},|\vec{a}|=1,|\vec{b}|=2|\vec{c}|=3$ then $\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}$ is equal to

• A. 0
• B. -7
• C. 7
• D. 1

Answer 1

Answer: (B)

$(\vec{a}+\vec{b}+\vec{c}{)}^2=(\vec{a}+\vec{b}+\vec{c})\cdot (\vec{a}+\vec{b}+\vec{c})$
$\Rightarrow (\vec{a}+\vec{b}+\vec{c}{)}^2=|\vec{a}{|}^2+|\vec{b}{|}^2+|\vec{c}{|}^2$
$+2(\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a})$
$\Rightarrow 0=1^2+2^2+3^2+2(\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a})$
$\Rightarrow 2(\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a})=-14$
$\Rightarrow \vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}=-7$