Let $\mathbf{u}$ and $\mathbf{v}$ be two vectors in $R^2$. If $|\mathbf{u}+\mathbf{v}|^2=2\left(|\mathbf{u}|^2+|\mathbf{v}|^2\right)$, then .....

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Aicharya · Accepted Answer

$ \begin{aligned} & 	ext { }|\mathbf{u}+\mathbf{v}|^2=2\left(|u|^2+|v|^2ight) \ &|\mathbf{u}|^2+|\mathbf{v}|^2+2 \mathbf{u v}=2|\mathbf{u}|^2+2|\mathbf{v}|^2 \ &|\mathbf{u}|^2+|\mathbf{v}|^2=2 \mathbf{u} \cdot \mathbf{v} \ &|\mathbf{u}|^2+|\mathbf{v}|^2-2 \mathbf{u} \cdot v=0 \ &(\mathbf{u}-\mathbf{v})^2=0 \ & \mathbf{u}-\mathbf{v}=0 \ & \mathbf{u}=\mathbf{v} \end{aligned} $ Hence, option (1) is correct

Let $\mathbf{u}$ and $\mathbf{v}$ be two vectors in $R^2$. If…

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