Ответ:

[tex] | \overrightarrow{a} - \overrightarrow{b}| = \sqrt{19} [/tex]

[tex] \sqrt{(\overrightarrow{a} - \overrightarrow{b})^{2} } = \sqrt{19} [/tex]

[tex] \sqrt{ {\overrightarrow{a}}^{2} - 2\overrightarrow{a}\overrightarrow{b} + {\overrightarrow{b}}^{2} } = \sqrt{19} [/tex]

Возведем обе части в квадрат

[tex]\overrightarrow{a}^{2} - 2\overrightarrow{a}\overrightarrow{b} + {\overrightarrow{b}}^{2} = 19[/tex]

Если что: [tex] {\overrightarrow{a}}^{2}= |{\overrightarrow{a}}|^{2} [/tex]

[tex] {3}^{2} - 2\overrightarrow{a}\overrightarrow{b} + {5}^{2} = 19 \\ 9 - 2\overrightarrow{a}\overrightarrow{b} + 25 = 19 \\ 34 - 2\overrightarrow{a}\overrightarrow{b} = 19 \\ - 2\overrightarrow{a}\overrightarrow{b} = 19 - 34 \\ - 2\overrightarrow{a}\overrightarrow{b} = - 15 \\ 2\overrightarrow{a}\overrightarrow{b} = 15[/tex]

[tex] |\overrightarrow{a} + \overrightarrow{b}| = \sqrt{( {\overrightarrow{a} + \overrightarrow{b})}^{2} } = \sqrt{ {\overrightarrow{a}}^{2} + 2\overrightarrow{a}\overrightarrow{b} +{\overrightarrow{b}}^{2} } [/tex]

[tex] \sqrt{ {3}^{2} + 15 + {5}^{2} } = \sqrt{9 + 15 + 25} = \sqrt{49} = 7[/tex]

Ответ:

[tex] |\overrightarrow{a} + \overrightarrow{b}| = 7[/tex]