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Added Cartesian coordinate analogy.
:<math> a \cdot \vec{v} = (a v_1,a v_2,a v_3)</math>
:<math> a \cdot \vec{v} = (a v_1,a v_2,a v_3)</math>
Because of this representation, each vector can be thought of as a point in a [[Cartesian coordinate system]]; or, rather, as an arrow from the [[origin]] to the point. For example, <math>\vec{v} = (4,5,6)</math> can be thought of as the arrow from the origin <tt>(0, 0, 0)</tt> to the point <tt>4,5,6</tt>. In this case, (4,5,6) would be called the ''head'' of ''v'', and (0,0,0) would be called the ''tail''.
Vector spaces are the fundamental objects of study of [[linear algebra]] and, due to their usefulness with [[gradient]]s, advanced calculus.
Vector spaces are the fundamental objects of study of [[linear algebra]] and, due to their usefulness with [[gradient]]s, advanced calculus.