a) $$\overrightarrow { p } \circ \overrightarrow { q } =-11\\ \overrightarrow { p } \circ \overrightarrow { q } =\left\| \overrightarrow { p } \right\| \left\| \overrightarrow { q } \right\| \cos { \varphi } =15\cos { \varphi } =-11\\ \cos { \varphi } =-\frac { 11 }{ 15 } ;\quad \varphi =\arccos { \left( -\frac { 11 }{ 15 } \right) } \approx 137,17°$$
$${ \overrightarrow { p } }_{ \overrightarrow { q } }=\frac { \overrightarrow { p } \circ \overrightarrow { q } }{ { \left\| \overrightarrow { q } \right\| }^{ 2 } } \overrightarrow { q } =\frac { -11 }{ 9 } \overrightarrow { q } \\ \left\| { \overrightarrow { p } }_{ \overrightarrow { q } } \right\| =\frac { 11 }{ 3 } $$
b) $$\overrightarrow { n } =\frac { \overrightarrow { p } \times \overrightarrow { q } }{ \left\| \overrightarrow { p } \times \overrightarrow { q } \right\| } =\frac { 1 }{ 2\sqrt { 26 } } { \left( -8,2,6 \right) }^{ T }$$
c) $$\overrightarrow { s } ={ \overrightarrow { p } +\overrightarrow { q } +\lambda \overrightarrow { n } =\left( 2,2,2 \right) }^{ T }+\lambda \overrightarrow { n } ={ \left( 2+\lambda { n }_{ 1 },2+\lambda { n }_{ 2 },2+\lambda { n }_{ 3 } \right) }^{ T }\\ \left\| \overrightarrow { s } \right\| =\sqrt { { \left( 2+\lambda { n }_{ 1 } \right) }^{ 2 }+{ \left( 2+\lambda { n }_{ 2 } \right) }^{ 2 }+{ \left( 2+\lambda { n }_{ 3 } \right) }^{ 2 } } =\sqrt { 13 } \\ \lambda =-1$$
d) $$A={ \left\| \overrightarrow { p } \times \overrightarrow { q } \right\| = }2\sqrt { 26 } \\ $$
e) $$V=\left\| \overrightarrow { p } \circ \left( \overrightarrow { q } \times \overrightarrow { n } \right) \right\| =2\sqrt { 26 } \\ $$
