Question: The definition \(\mathbf{u}\cdot \mathbf{v}=||\mathbf{u}||||\mathbf{v}||cos\theta \) implies that \(|\mathbf{u}\cdot \mathbf{v}|\le ||\mathbf{u}||||\mathbf{v}||\). This is called the Cauchy-Schwarz Inequality.

1. Show that \(||\mathbf{u}+\mathbf{v}|{{|}^{2}}=|\mathbf{u}|{{|}^{2}}+2\mathbf{u}\cdot \mathbf{v}+||\mathbf{v}|{{|}^{2}}\)
2. Use the Cauchy-Schwarz Inequality to show that \(||\mathbf{u}+\mathbf{v}|{{|}^{2}}\le {{(||\mathbf{u}||+||\mathbf{v}||)}^{2}}\).
3. Conclude that \(\,\,||\mathbf{u}+\mathbf{v}||\,\le \,||\mathbf{u}||+||\mathbf{v}||\) This is called the Triangle Inequality.

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