An orthonormal basis that contains a positive scalar multiple of [ 3 , 1 , 5 ] {\displaystyle [3,1,5]} is
[ 3 35 , 1 35 , 5 35 ] , [ − 5 34 , 0 , 3 34 ] , [ 3 1190 , − 34 1190 , 5 1190 ] {\displaystyle [{\frac {3}{\sqrt {35}}},{\frac {1}{\sqrt {35}}},{\frac {5}{\sqrt {35}}}],[{\frac {-5}{\sqrt {34}}},0,{\frac {3}{\sqrt {34}}}],[{\frac {3}{\sqrt {1190}}},{\frac {-34}{\sqrt {1190}}},{\frac {5}{\sqrt {1190}}}]}
The corresponding orthogonal matrix is
( 3 35 − 5 34 3 1190 1 35 0 − 34 1190 5 35 3 34 5 1190 ) {\displaystyle {\begin{pmatrix}{\frac {3}{\sqrt {35}}}&{\frac {-5}{\sqrt {34}}}&{\frac {3}{\sqrt {1190}}}\\{\frac {1}{\sqrt {35}}}&0&{\frac {-34}{\sqrt {1190}}}\\{\frac {5}{\sqrt {35}}}&{\frac {3}{\sqrt {34}}}&{\frac {5}{\sqrt {1190}}}\end{pmatrix}}}
The angle between the vectors [ 1 , 0 , 0 ] {\displaystyle [1,0,0]} and [ 1 , 1 , 0 ] {\displaystyle [1,1,0]} is π 4 {\displaystyle {\frac {\pi }{4}}} , so the angle between [ 3 , 1 , 5 ] {\displaystyle [3,1,5]} and the following vector is π 4 {\displaystyle {\frac {\pi }{4}}}
( 3 35 − 5 34 3 1190 1 35 0 − 34 1190 5 35 3 34 5 1190 ) ( 1 1 0 ) = ( 3 35 − 5 34 1 35 5 35 + 3 34 ) {\displaystyle {\begin{pmatrix}{\frac {3}{\sqrt {35}}}&{\frac {-5}{\sqrt {34}}}&{\frac {3}{\sqrt {1190}}}\\{\frac {1}{\sqrt {35}}}&0&{\frac {-34}{\sqrt {1190}}}\\{\frac {5}{\sqrt {35}}}&{\frac {3}{\sqrt {34}}}&{\frac {5}{\sqrt {1190}}}\end{pmatrix}}{\begin{pmatrix}1\\1\\0\end{pmatrix}}={\begin{pmatrix}{\frac {3}{\sqrt {35}}}-{\frac {5}{\sqrt {34}}}\\{\frac {1}{\sqrt {35}}}\\{\frac {5}{\sqrt {35}}}+{\frac {3}{\sqrt {34}}}\end{pmatrix}}}
A n s w e r : ( 3 35 − 5 34 1 35 5 35 + 3 34 ) {\displaystyle \color {blue}Answer:{\begin{pmatrix}{\frac {3}{\sqrt {35}}}-{\frac {5}{\sqrt {34}}}\\{\frac {1}{\sqrt {35}}}\\{\frac {5}{\sqrt {35}}}+{\frac {3}{\sqrt {34}}}\end{pmatrix}}}
![{\displaystyle [3,1,5]}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/7e7a46755ab2bc7a351265b5696167b9fe5bf404)
is
![{\displaystyle [{\frac {3}{\sqrt {35}}},{\frac {1}{\sqrt {35}}},{\frac {5}{\sqrt {35}}}],[{\frac {-5}{\sqrt {34}}},0,{\frac {3}{\sqrt {34}}}],[{\frac {3}{\sqrt {1190}}},{\frac {-34}{\sqrt {1190}}},{\frac {5}{\sqrt {1190}}}]}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/7b471817f53d1ef19ff7807be4b1da504518a96b)
![{\displaystyle [1,0,0]}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/34bebbc071daa488f734743ef25b5a73c8ed1450)
and ![{\displaystyle [1,1,0]}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/374b112ab3da934db01e82a78c3a947a38a13066)
is 
, so the angle between and the following vector is
