# Vector

Vector is a directed line segment. A vector $\vec{a} = (x_a, \, y_a, \, z_a)$ has the magnitude $|\vec{a}| = \sqrt{x_a^2 + y_a^2 + z_a^2}$ . If the magnitude is equal $1$ , the vector is called a unit vector and denoted by a hat (i.e. $\hat{a} = (\frac{3}{5}, \, 0, \, \frac{4}{5})$ ). If all components are equal $0$ , it is a zero vector, which has no direction and its magnitude is $0$ . Special unit vectors:
$$\begin{array}{l} \vec{i} = (1, \, 0, \, 0) \qquad \vec{j} = (0, \, 1, \, 0) \qquad \vec{k} = (0, \, 0, \, 1) \end{array}$$
$$\begin{array}{l} t\vec{a} = (tx_a, \, ty_a, \, tz_a) \qquad |t\vec{a}| = |t||\vec{a}| \qquad \vec{a} \parallel \vec{b} \, \Leftrightarrow \, \vec{b} = t\vec{a} \qquad \hat{a} = |a|^{-1} \vec{a} \\[6pt] \forall \, \vec{a} = (x_a, \, y_a, \, z_a) = x_a \vec{i} + y_a \vec{j} + z_a \vec{k} \\[6pt] \vec{a} + \vec{b} = (x_a + x_b, \, y_a + y_b, \, z_a + z_b) \end{array}$$