  A vector \(A\), written ‘\(\ket{A}\)’, is a mathematical object characterized by a length, \(|A|\), and a direction. A normalized vector is a vector of length 1; i.e., \(|A| = 1\). Vectors can be added together, multiplied by constants (including complex numbers), and multiplied together. Vector addition maps any pair of vectors onto another vector, specifically, the one you get by moving the second vector so that its tail coincides with the tip of the first, without altering the length or direction of either, and then joining the tail of the first to the tip of the second. This addition rule is known as the parallelogram law. So, for example, adding vectors \(\ket{A}\) and \(\ket{B}\) yields vector \(\ket{C} (= \ket{A} + \ket{B})\) as in Figure 1: 