arrow_back If $\mathbf{u}$ and $\mathbf{v}$ are orthogonal vectors in $R^{n}$ with the Euclidean inner product, then prove that $$\|\mathbf{u}+\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}$$

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If $\mathbf{u}$ and $\mathbf{v}$ are orthogonal vectors in $R^{n}$ with the Euclidean inner product, then prove that $$\|\mathbf{u}+\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}$$

Proof

Since $\mathbf{u}$ and $\mathbf{v}$ are orthogonal, we have $\mathbf{u} \cdot \mathbf{v}=0$, from which it follows that
$$\|\mathbf{u}+\mathbf{v}\|^{2}=(\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=\|\mathbf{u}\|^{2}+2(\mathbf{u} \cdot \mathbf{v})+\|\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}$$

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