If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are three vectors such that $|\mathbf{a}|=3,|\mathbf{b}|=4$ and $|\mathbf{c}|=5$ and $\mathbf{a}+\mathbf{b}+\mathbf{c}=0$, then $\mathbf{a} \cdot \mathbf{b}$ is equal to

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Aicharya · Accepted Answer

$\begin{aligned} & 	ext { } \mathbf{a}, \mathbf{b}, \mathbf{c} 	ext { are three vectors such that }|\mathbf{a}|=3 	ext {, } \ & \begin{array}{l}|\mathbf{b}|=4,|\mathbf{c}|=5 	ext { and } \mathbf{a}+\mathbf{b}+\mathbf{c}=0 \ 	ext { Now, } \mathbf{a}+\mathbf{b}+\mathbf{c}=0 \ \quad \mathbf{a}+\mathbf{b}=-\mathbf{c}\end{array} \ & \begin{aligned} \Rightarrow \quad(\mathbf{a}+\mathbf{b})^2=(-\mathbf{c})^2\end{aligned} \ & \begin{aligned} & \Rightarrow|\mathbf{a}|^2+|\mathbf{b}|^2+2 \mathbf{a} \cdot \mathbf{b}=|\mathbf{c}|^2\left[\because \mathbf{a} \cdot \mathbf{a}=|\mathbf{a}|^2ight] \ & \Rightarrow 3^2+4^2+2 \mathbf{a} \cdot \mathbf{b}=5^2 \ & \Rightarrow \quad 2 \mathbf{a} \cdot \mathbf{b}=25-9-16 \ & \Rightarrow \quad 2 \mathbf{a} \cdot \mathbf{b}=0 \ & \Rightarrow \mathbf{a} \cdot \mathbf{b}=0\end{aligned}\end{aligned}$

If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are three vectors such that $|\mathbf{a}|=3,|\mathbf{b}|=4$ and…

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