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

    \(\left\| \mathbf u + \mathbf v \right\| = 13\), \(\left\| \mathbf u - \mathbf v \right\| = 13\).

Difficulty level: Moderate task
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