15. (a) Find 3 -dimensional vectors $\overrightarrow { v _ { 1 } } , \overrightarrow { v _ { 2 } } , \overrightarrow { v _ { 3 } }$ satisfying
$$\overrightarrow { v _ { 1 } } \cdot \overrightarrow { v _ { 1 } } = 4 , \overrightarrow { v _ { 1 } } \cdot \overrightarrow { v _ { 2 } } = - 2 , \overrightarrow { v _ { 1 } } \cdot \overrightarrow { v _ { 3 } } = 6 , \overrightarrow { v _ { 2 } } \cdot \overrightarrow { v _ { 2 } } = 2 , \overrightarrow { v _ { 2 } } \cdot \overrightarrow { v _ { 3 } } = - 5 , \overrightarrow { v _ { 3 } } \cdot \overrightarrow { v _ { 3 } } = 29 .$$
(b) Let $\vec { A } ( t ) = f _ { 1 } ( t ) \hat { i } + f _ { 2 } ( t ) \hat { j }$ and $B ( t ) = g _ { 1 } ( t ) \hat { i } + g _ { 2 } ( t ) \hat { j } , t \in [ 0,1 ]$, where $f _ { 1 \text {, } } \mathrm { f } _ { 2 } , \mathrm {~g} _ { 1 } , \mathrm {~g} _ { 2 }$ are continuous functions. If $\vec { A } ( \mathrm { t } )$ and $\vec { B } ( \mathrm { t } )$ are non-zero vectors for all t and $\vec { A } ( 0 ) = 2 \hat { \imath } + 3 \hat { \jmath } , \vec { A } ( 1 ) = 6 \hat { \imath } + 2 \hat { \jmath } , \vec { B } ( 0 ) = 3 \hat { \imath } + 2 \hat { \jmath }$ and $\vec { B } ( 1 ) = 2 \hat { \imath } + 6 \hat { j }$. The show that $\vec { A } ( t )$ and $\vec { B } ( t )$ are parallel for some $t$.