Vectors

Abstract

We have defined a 1 × n matrix as a row vector and an n × 1 matrix as a column vector. A particularly important application of matrix algebra comes when n = 3, and we are dealing with things that happen in the three-dimensional, real space we live in. A vector is then a quantity with a magnitude and a direction. Let x be the column vector

$$x = \left( {\begin{array}{*{20}{c}} {{{x}_{1}}} \\ {{{x}_{2}}} \\ {{{x}_{3}}} \\ \end{array} } \right)$$

We define its magnitude by $|x| = \sqrt {{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}}} = {{({{x}^{T}}x)}^{{1/2}}}$, and consider a line whose direction cosines, with respect to an orthonormal system are given by cos α _i = x _i /|x|. We can reach any point in this three-dimensional space by moving a distance x₁ parallel to axis 1, a distance x₂ parallel to axis 2, and a distance x₃ parallel to axis 3. If we multiply the vector x by the inversion matrix,

$$R\left( {\begin{array}{*{20}{c}} { - 1} & 0 & 0 \\ 0 & { - 1} & 0 \\ 0 & 0 & { - 1} \\ \end{array} } \right),$$

we get the vector −x, which corresponds to a displacement in the opposite direction from the origin. If we apply forces to a particle proportional to x₁, x₂ and x₃ parallel to the 1, 2, and 3 axes, respectively, the acceleration will be in the direction of, and proportional to, x, and, again, a force in the opposite direction will produce an acceleration in the opposite direction. Displacements, linear velocities, forces, and accelerations are examples of polar vectors, and they are characterized by this property of reversing direction as a result of a space inversion.