\r\n Consider a 2D Cartesian coordinate system x-y defined by a set of mutually orthogonal unit vectors , which are also known as basis vectors of the coordinate system.\r\n A vector, say \\(\\mathbf r\\), in such a coordinate system can be represented by means of a doublet of scalars:\r\n $$\\mathbf r = r_x \\mathbf x + r_y \\mathbf y \\tag{1}$$\r\n The scalars \\(r_x\\) and \\(r_y\\) are called the components of vector \\(\\mathbf r\\) in the Cartesian basis defined by unit vectors \\(\\mathbf x\\) and \\(\\mathbf y\\).\r\n Upon taking the scalar product of \\(\\mathbf x\\) and \\(\\mathbf r\\), we obtain:\r\n $$\\mathbf r \\cdot \\mathbf x = r_x \\mathbf x \\cdot \\mathbf x + r_y \\mathbf y \\cdot \\mathbf x \\tag{2}$$\r\n Note that the basis vectors \\(\\mathbf x \\) and \\(\\mathbf y \\)are mutually orthogonal, i.e.,\r\n $$ \\mathbf x \\cdot \\mathbf y = \\mathbf y \\cdot \\mathbf z = \\mathbf z \\cdot \\mathbf x = 0 \\tag{3}$$\r\n Further, since the basis vectors are unit vectors, we obtain:\r\n $$ \\mathbf x \\cdot \\mathbf x = \\mathbf y \\cdot \\mathbf y = \\mathbf z \\cdot \\mathbf z =1 \\tag{4}$$\r\n Using the results from equations (2) and (3) in equation (4), we obtain\r\n $$r_x = \\mathbf r \\cdot \\mathbf x \\tag{5}$$\r\n Following a similar process, we can show that:\r\n $$r_y = \\mathbf r \\cdot \\mathbf y \\tag{6}$$\r\n Note that although the vector \\(\\mathbf r \\) is independent of the choice of coordinate system, however, the components \\(r_x\\), and \\(r_y\\) are not.\r\n
\r\n\r\n Now, consider another Two-D Cartesian coordinate system x'-y' which is rotated by an angle \\(\\theta \\) w.r.t. original coordinate system x-y, as shown in the figure below.\r\n
\r\n\r\n The rotated coordinate system is defined by basis vectors \\(\\mathbf x'\\) and \\(\\mathbf y'\\) which are related to original basis vectors as follows:\r\n $$\\mathbf x' = \\cos \\theta\\ \\mathbf x + \\sin \\theta\\ \\mathbf y \\tag{7}$$\r\n and\r\n $$\\mathbf y' = -\\sin \\theta \\ \\mathbf x + \\cos \\theta \\ \\mathbf y \\tag{8}$$\r\n Let's say the components of vector \\(\\mathbf r\\) in the new Cartesian basis\r\n $$\\mathbf r = r_x' \\mathbf x' + r_y' \\mathbf y' \\tag{5}$$\r\n Taking the dot product with \\(\\mathbf x'\\) gives\r\n $$ r_{x'} = \\mathbf r \\cdot \\mathbf x' = r_x \\mathbf x \\cdot \\mathbf x' + r_y \\mathbf y \\cdot \\mathbf x'= r_x \\cos \\theta + r_y \\sin \\theta \\tag{9}$$\r\n Upon taking the dot product with \\(\\mathbf y'\\) gives\r\n $$ r_{y'} = \\mathbf r \\cdot \\mathbf y' = r_x \\mathbf x \\cdot \\mathbf y' + r_y \\mathbf y \\cdot \\mathbf y'= -r_x \\sin \\theta + r_y \\cos \\theta \\tag{10}$$\r\n
\r\n\r\n In tensorial notation, the above transformation law can be written as:\r\n $$\\begin{bmatrix} r_{x'} \\\\ r_{y'} \\end{bmatrix} = \\begin{bmatrix} \\cos \\theta & \\sin \\theta \\\\ -\\sin \\theta & \\cos \\theta \\end{bmatrix}\\cdot \\begin{bmatrix} r_{x} \\\\ r_{y} \\end{bmatrix}$$\r\n
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