## Norm of the sum

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

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