Trong không gian với hệ tọa độ Oxyz (Hình 36), cho hai vectơ
\(\overrightarrow{u}=\left(x_1;y_1;z_1\right)\) và \(\overrightarrow{v}=\left(x_2;y_2;z_2\right)\).
a) Biểu diễn các vectơ \(\overrightarrow{u},\overrightarrow{v}\) theo ba vectơ \(\overrightarrow{i},\overrightarrow{j},\overrightarrow{k}.\)
b) Biểu diễn các vectơ \(\overrightarrow{u}+\overrightarrow{v},\overrightarrow{u}-\overrightarrow{v},m\overrightarrow{u}\left(m\in\text{ℝ}\right)\) theo ba vectơ \(\overrightarrow{i},\overrightarrow{j},\overrightarrow{k}.\)
c) Tìm tọa độ của các vectơ \(\overrightarrow{u}+\overrightarrow{v},\overrightarrow{u}-\overrightarrow{v},m\overrightarrow{u}\left(m\in\text{ℝ}\right)\)
a) \(\overrightarrow u = ({x_1};{y_1};{z_1}) = {x_1}\overrightarrow i + {y_1}\overrightarrow j + {z_1}\overrightarrow k \).
\(\overrightarrow v = ({x_2};{y_2};{z_2}) = {x_2}\overrightarrow i + {y_2}\overrightarrow j + {z_2}\overrightarrow k \).
b) \(\overrightarrow u + \overrightarrow v = {x_1}\overrightarrow i + {y_1}\overrightarrow j + {z_1}\overrightarrow k + {x_2}\overrightarrow i + {y_2}\overrightarrow j + {z_2}\overrightarrow k = ({x_1} + {x_2})\overrightarrow i + ({y_1} + {y_2})\overrightarrow j + ({z_1} + {z_2})\overrightarrow k \).
\(\overrightarrow u - \overrightarrow v = {x_1}\overrightarrow i + {y_1}\overrightarrow j + {z_1}\overrightarrow k - {x_2}\overrightarrow i - {y_2}\overrightarrow j - {z_2}\overrightarrow k = ({x_1} - {x_2})\overrightarrow i + ({y_1} - {y_2})\overrightarrow j + ({z_1} - {z_2})\overrightarrow k \).
\(m\overrightarrow u = m({x_1}\overrightarrow i + {y_1}\overrightarrow j + {z_1}\overrightarrow k ) = m{x_1}\overrightarrow i + m{y_1}\overrightarrow j + m{z_1}\overrightarrow k \).
c) \(\overrightarrow u + \overrightarrow v = ({x_1} + {x_2})\overrightarrow i + ({y_1} + {y_2})\overrightarrow j + ({z_1} + {z_2})\overrightarrow k = ({x_1} + {x_2};{y_1} + {y_2};{z_1} + {z_2})\).
\(\overrightarrow u - \overrightarrow v = ({x_1} - {x_2})\overrightarrow i + ({y_1} - {y_2})\overrightarrow j + ({z_1} - {z_2})\overrightarrow k = ({x_1} - {x_2};{y_1} - {y_2};{z_1} - {z_2})\).
\(m\overrightarrow u = m{x_1}\overrightarrow i + m{y_1}\overrightarrow j + m{z_1}\overrightarrow k = (m{x_1};m{y_1};m{z_1})\).