{"id":5241,"date":"2014-04-23T06:05:01","date_gmt":"2014-04-22T21:05:01","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5241"},"modified":"2014-08-01T18:30:24","modified_gmt":"2014-08-01T09:30:24","slug":"orthogonal-vectors","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5241","title":{"rendered":"Orthogonal vectors"},"content":{"rendered":"
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The scalar or dot product of two vectors a_1\\bold{i} + a_2\\bold{j} + a_3\\bold{k} and b_1\\bold{i} + b_2\\bold{j} + b_3\\bold{k} is a_1b_1 + a_2b_2 + a_3b_3 and the vectors are perpendicular or orthogonal if a_1b_1 + a_2b_2 + a_3b_3 = 0. From the point of view of matrices we can consider these vectors as column vectors <\/p>\n

\\displaystyle A = \\left( \\begin{array}{c} a_1 \\\\ a_2 \\\\ a_3 \\end{array} \\right),\\ B = \\left( \\begin{array}{c} b_1 \\\\ b_2 \\\\ b_3 \\end{array} \\right)<\/p>\n

from which it follows that A^TB = a_1b_1 + a_2b_2 + a_3b_3. <\/p>\n

This leads us to define the scalar product of real column vectors<\/em> A and B<\/latex> as A^TB and to define A and B to be orthogonal<\/em> if A^TB = 0. <\/p>\n

It is convenient to generalize this to cases where the vectors can have complex components and we adopt the following definition: <\/p>\n

Definition 1.<\/strong> Two column vectors A and B are called orthogonal<\/em> if \\bar{A}^TB = 0 , and \\bar{A}^TB is called the scalar product<\/em> of A and B. <\/p>\n

It should be noted also that if A is a unitary matrix then \\bar{A}^TA = 1, which means that the scalar product of A with itself is 1 or equivalently A is a unit vector<\/em>, i.e. having length 1. Thus a unitary column vector is a unit vector. Because of these remarks we have the following <\/p>\n

Definition 2.<\/strong> A set of vectors X_1,\\ X_2,\\ \\cdots for which <\/p>\n

\\displaystyle \\bar{X}^T_jX_k = \\left\\{\\begin{array}{cc} 0 & j \\ne k \\\\ 1 & j = k \\end{array} \\right.<\/p>\n

is called a unitary set or system of vectors<\/em> or, in the case where the vectors are real, an orthonormal set<\/em> or an orthogonal set of unit vectors<\/em>. <\/p>\n<\/blockquote>\n