Norm of the sum
Task number: 2694
Let be given two perpendicular vectors \(\mathbf u\) and \(\mathbf v\) s.t. \(\left\| \mathbf u \right\| = 12, \left\| \mathbf v \right\| = 5\). Determine \(\left\| \mathbf u + \mathbf v \right\|\) and \(\left\| \mathbf u - \mathbf v \right\|\).
Hint
Orthogonality means that \(\langle \mathbf u|\mathbf v \rangle=0\). Then use the definition of the norm and axioms of the scalar product.
Resolution
From norms we get \(\langle \mathbf u|\mathbf u \rangle=144\) and \(\langle \mathbf v|\mathbf v \rangle=25\).
Then \( \left\| \mathbf u + \mathbf v \right\| = \sqrt{ \langle \mathbf u+\mathbf v|\mathbf u+\mathbf v \rangle } = \sqrt{ \langle \mathbf u|\mathbf u+\mathbf v \rangle + \langle \mathbf v |\mathbf u+\mathbf v \rangle } = \sqrt{ \overline{\langle \mathbf u+\mathbf v|\mathbf u \rangle} + \overline{\langle \mathbf u+\mathbf v|\mathbf v \rangle}} = \)
\( \sqrt{ \overline{\langle \mathbf u|\mathbf u \rangle} + \overline{\langle \mathbf v|\mathbf u \rangle} + \overline{\langle \mathbf u|\mathbf v \rangle} + \overline{\langle \mathbf v|\mathbf v \rangle} } = \sqrt{ 144 + 25 } = 13 \).Result
\(\left\| \mathbf u + \mathbf v \right\| = 13\), \(\left\| \mathbf u - \mathbf v \right\| = 13\).
