{"id":2566,"date":"2025-08-13T18:07:24","date_gmt":"2025-08-13T18:07:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2566"},"modified":"2025-10-22T23:17:08","modified_gmt":"2025-10-22T23:17:08","slug":"rotation-of-axes-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rotation-of-axes-learn-it-2\/","title":{"raw":"Rotation of Axes: Learn It 2","rendered":"Rotation of Axes: Learn It 2"},"content":{"raw":"

Finding a New Representation of the Given Equation after Rotating through a Given Angle<\/h2>\r\nUntil now, we have looked at equations of conic sections without an [latex]xy[\/latex] term, which aligns the graphs with the x<\/em>- and y<\/em>-axes. When we add an [latex]xy[\/latex] term, we are rotating the conic about the origin. If the x<\/em>- and y<\/em>-axes are rotated through an angle, say [latex]\\theta [\/latex], then every point on the plane may be thought of as having two representations: [latex]\\left(x,y\\right)[\/latex] on the Cartesian plane with the original x<\/em>-axis and y<\/em>-axis, and [latex]\\begin{align}\\left({x}^{\\prime },{y}^{\\prime }\\right)\\end{align}[\/latex] on the new plane defined by the new, rotated axes, called the x'<\/em>-axis and y'<\/em>-axis.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\" The graph of the rotated ellipse [latex]{x}^{2}+{y}^{2}-xy - 15=0[\/latex][\/caption]We will find the relationships between [latex]x[\/latex] and [latex]y[\/latex] on the Cartesian plane with [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] on the new rotated plane.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\" The Cartesian plane with x- and y-axes and the resulting x\u2032\u2212 and y\u2032\u2212axes formed by a rotation by an angle [latex]\\theta [\/latex].[\/caption]The original coordinate x<\/em>- and y<\/em>-axes have unit vectors [latex]i[\/latex] and [latex]j[\/latex]. The rotated coordinate axes have unit vectors [latex]\\begin{align}{i}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{j}^{\\prime }\\end{align}[\/latex]. The angle [latex]\\theta [\/latex] is known as the angle of rotation<\/strong>. We may write the new unit vectors in terms of the original ones.\r\n
[latex]\\begin{align}&{i}^{\\prime }=i\\cos \\theta +j\\sin \\theta \\\\ &{j}^{\\prime }=-i\\sin \\theta +j\\cos \\theta \\end{align}[\/latex]<\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\" Relationship between the old and new coordinate planes.[\/caption]\r\n\r\nConsider a vector [latex]u[\/latex] <\/strong>in the new coordinate plane. It may be represented in terms of its coordinate axes.\r\n
[latex]\\begin{align}&u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime } \\\\ &u={x}^{\\prime }\\left(i\\cos \\theta +j\\sin \\theta \\right)+{y}^{\\prime }\\left(-i\\sin \\theta +j\\cos \\theta \\right) && \\text{Substitute}. \\\\ &u=ix^{\\prime}\\cos \\theta +jx^{\\prime}\\sin \\theta -iy^{\\prime}\\sin \\theta +jy^{\\prime}\\cos \\theta && \\text{Distribute}. \\\\ &u=ix^{\\prime}\\cos \\theta -iy^{\\prime}\\sin \\theta +jx^{\\prime}\\sin \\theta +jy^{\\prime}\\cos \\theta && \\text{Apply commutative property}. \\\\ &u=\\left(x^{\\prime}\\cos \\theta -y^{\\prime}\\sin \\theta \\right)i+\\left(x^{\\prime}\\sin \\theta +y^{\\prime}\\cos \\theta \\right)j && \\text{Factor by grouping}. \\end{align}[\/latex]<\/div>\r\nBecause [latex]\\begin{align}u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime }\\end{align}[\/latex], we have representations of [latex]x[\/latex] and [latex]y[\/latex] in terms of the new coordinate system.\r\n
[latex]\\begin{gathered}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ \\text{and}\\\\ y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{gathered}[\/latex]<\/div>\r\n
\r\n

equations of rotation<\/h3>\r\nIf a point [latex]\\left(x,y\\right)[\/latex] on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle [latex]\\theta [\/latex] from the positive x<\/em>-axis, then the coordinates of the point with respect to the new axes are [latex]\\begin{align}\\left({x}^{\\prime },{y}^{\\prime }\\right)\\end{align}[\/latex]. We can use the following equations of rotation to define the relationship between [latex]\\begin{align}\\left(x,y\\right)\\end{align}[\/latex] and [latex]\\begin{align}\\left({x}^{\\prime },{y}^{\\prime }\\right):\\end{align}[\/latex]\r\n

[latex]\\begin{gathered}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ \\text{and}\\\\ y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{gathered}[\/latex]<\/p>\r\n\r\n<\/section><\/div>\r\n

How To: Given the equation of a conic, find a new representation after rotating through an angle.\r\n<\/strong>\r\n
    \r\n \t
  1. Find [latex]x[\/latex] and [latex]y[\/latex] where [latex]\\begin{align}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\end{align}[\/latex] and [latex]\\begin{align}y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/latex].<\/li>\r\n \t
  2. Substitute the expression for [latex]x[\/latex] and [latex]y[\/latex] into in the given equation, then simplify.<\/li>\r\n \t
  3. Write the equations with [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] in standard form.<\/li>\r\n<\/ol>\r\n<\/section>
    Find a new representation of the equation [latex]2{x}^{2}-xy+2{y}^{2}-30=0[\/latex] after rotating through an angle of [latex]\\theta =45^\\circ [\/latex].[reveal-answer q=\"303707\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"303707\"]Find [latex]x[\/latex] and [latex]y[\/latex], where [latex]\\begin{align}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\end{align}[\/latex] and [latex]\\begin{align} y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/latex].\r\n\r\nBecause [latex]\\theta =45^\\circ [\/latex],\r\n

    [latex]\\begin{align} &x={x}^{\\prime }\\cos \\left(45^\\circ \\right)-{y}^{\\prime }\\sin \\left(45^\\circ \\right) \\\\ &x={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)-{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right) \\\\ &x=\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}} \\end{align}[\/latex]<\/p>\r\nand\r\n

    [latex]\\begin{align}&y={x}^{\\prime }\\sin \\left(45^\\circ \\right)+{y}^{\\prime }\\cos \\left(45^\\circ \\right) \\\\ &y={x}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right)+{y}^{\\prime }\\left(\\frac{1}{\\sqrt{2}}\\right) \\\\ &y=\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}} \\end{align}[\/latex]<\/p>\r\nSubstitute [latex]\\begin{align}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\end{align}[\/latex] and [latex]\\begin{align}y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/latex] into [latex]2{x}^{2}-xy+2{y}^{2}-30=0[\/latex].\r\n

    [latex]\\begin{align} 2{\\left(\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}}\\right)}^{2}-\\left(\\frac{{x}^{\\prime }-{y}^{\\prime }}{\\sqrt{2}}\\right)\\left(\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}}\\right)+2{\\left(\\frac{{x}^{\\prime }+{y}^{\\prime }}{\\sqrt{2}}\\right)}^{2}-30=0\\end{align}[\/latex]<\/p>\r\nSimplify.\r\n

    [latex]\\begin{align}&2\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }-{y}^{\\prime }\\right)}{2}-\\frac{\\left({x}^{\\prime }-{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{2}+2\\frac{\\left({x}^{\\prime }+{y}^{\\prime }\\right)\\left({x}^{\\prime }+{y}^{\\prime }\\right)}{2}-30=0 && \\text{FOIL method} \\\\ &{x}^{\\prime }{}^{2}{-2{x}^{\\prime }y}^{\\prime }+{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}+{x}^{\\prime }{}^{2}+2{x}^{\\prime }{y}^{\\prime }+{y}^{\\prime }{}^{2}-30=0 && \\text{Combine like terms}. \\\\ &2{x}^{\\prime }{}^{2}+2{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}=30 && \\text{Combine like terms}. \\\\ &2\\left(2{x}^{\\prime }{}^{2}+2{y}^{\\prime }{}^{2}-\\frac{\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)}{2}\\right)=2\\left(30\\right) && \\text{Multiply both sides by 2}. \\\\ &4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-\\left({x}^{\\prime }{}^{2}-{y}^{\\prime }{}^{2}\\right)=60 && \\text{Simplify}. \\\\ &4{x}^{\\prime }{}^{2}+4{y}^{\\prime }{}^{2}-{x}^{\\prime }{}^{2}+{y}^{\\prime }{}^{2}=60 && \\text{Distribute}. \\\\ &\\frac{3{x}^{\\prime }{}^{2}}{60}+\\frac{5{y}^{\\prime }{}^{2}}{60}=\\frac{60}{60} && \\text{Set equal to 1}. \\end{align}[\/latex]<\/p>\r\nWrite the equations with [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] in the standard form.\r\n

    [latex]\\begin{align}\\frac{{{x}^{\\prime }}^{2}}{20}+\\frac{{{y}^{\\prime }}^{2}}{12}=1\\end{align}[\/latex]<\/p>\r\nThis equation is an ellipse. Figure 6\u00a0shows the graph.\r\n\r\n\"\"\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

    Finding a New Representation of the Given Equation after Rotating through a Given Angle<\/h2>\n

    Until now, we have looked at equations of conic sections without an [latex]xy[\/latex] term, which aligns the graphs with the x<\/em>– and y<\/em>-axes. When we add an [latex]xy[\/latex] term, we are rotating the conic about the origin. If the x<\/em>– and y<\/em>-axes are rotated through an angle, say [latex]\\theta[\/latex], then every point on the plane may be thought of as having two representations: [latex]\\left(x,y\\right)[\/latex] on the Cartesian plane with the original x<\/em>-axis and y<\/em>-axis, and [latex]\\begin{align}\\left({x}^{\\prime },{y}^{\\prime }\\right)\\end{align}[\/latex] on the new plane defined by the new, rotated axes, called the x’<\/em>-axis and y’<\/em>-axis.<\/p>\n

    \"\"
    The graph of the rotated ellipse [latex]{x}^{2}+{y}^{2}-xy - 15=0[\/latex]<\/figcaption><\/figure>\n

    We will find the relationships between [latex]x[\/latex] and [latex]y[\/latex] on the Cartesian plane with [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] on the new rotated plane.<\/p>\n

    \"\"
    The Cartesian plane with x- and y-axes and the resulting x\u2032\u2212 and y\u2032\u2212axes formed by a rotation by an angle [latex]\\theta [\/latex].<\/figcaption><\/figure>\n

    The original coordinate x<\/em>– and y<\/em>-axes have unit vectors [latex]i[\/latex] and [latex]j[\/latex]. The rotated coordinate axes have unit vectors [latex]\\begin{align}{i}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{j}^{\\prime }\\end{align}[\/latex]. The angle [latex]\\theta[\/latex] is known as the angle of rotation<\/strong>. We may write the new unit vectors in terms of the original ones.<\/p>\n

    [latex]\\begin{align}&{i}^{\\prime }=i\\cos \\theta +j\\sin \\theta \\\\ &{j}^{\\prime }=-i\\sin \\theta +j\\cos \\theta \\end{align}[\/latex]<\/div>\n
    \"\"
    Relationship between the old and new coordinate planes.<\/figcaption><\/figure>\n

    Consider a vector [latex]u[\/latex] <\/strong>in the new coordinate plane. It may be represented in terms of its coordinate axes.<\/p>\n

    [latex]\\begin{align}&u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime } \\\\ &u={x}^{\\prime }\\left(i\\cos \\theta +j\\sin \\theta \\right)+{y}^{\\prime }\\left(-i\\sin \\theta +j\\cos \\theta \\right) && \\text{Substitute}. \\\\ &u=ix^{\\prime}\\cos \\theta +jx^{\\prime}\\sin \\theta -iy^{\\prime}\\sin \\theta +jy^{\\prime}\\cos \\theta && \\text{Distribute}. \\\\ &u=ix^{\\prime}\\cos \\theta -iy^{\\prime}\\sin \\theta +jx^{\\prime}\\sin \\theta +jy^{\\prime}\\cos \\theta && \\text{Apply commutative property}. \\\\ &u=\\left(x^{\\prime}\\cos \\theta -y^{\\prime}\\sin \\theta \\right)i+\\left(x^{\\prime}\\sin \\theta +y^{\\prime}\\cos \\theta \\right)j && \\text{Factor by grouping}. \\end{align}[\/latex]<\/div>\n

    Because [latex]\\begin{align}u={x}^{\\prime }{i}^{\\prime }+{y}^{\\prime }{j}^{\\prime }\\end{align}[\/latex], we have representations of [latex]x[\/latex] and [latex]y[\/latex] in terms of the new coordinate system.<\/p>\n

    [latex]\\begin{gathered}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ \\text{and}\\\\ y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{gathered}[\/latex]<\/div>\n
    \n
    \n

    equations of rotation<\/h3>\n

    If a point [latex]\\left(x,y\\right)[\/latex] on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle [latex]\\theta[\/latex] from the positive x<\/em>-axis, then the coordinates of the point with respect to the new axes are [latex]\\begin{align}\\left({x}^{\\prime },{y}^{\\prime }\\right)\\end{align}[\/latex]. We can use the following equations of rotation to define the relationship between [latex]\\begin{align}\\left(x,y\\right)\\end{align}[\/latex] and [latex]\\begin{align}\\left({x}^{\\prime },{y}^{\\prime }\\right):\\end{align}[\/latex]<\/p>\n

    [latex]\\begin{gathered}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\\\ \\text{and}\\\\ y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{gathered}[\/latex]<\/p>\n<\/section>\n<\/div>\n

    How To: Given the equation of a conic, find a new representation after rotating through an angle.
    \n<\/strong><\/p>\n
      \n
    1. Find [latex]x[\/latex] and [latex]y[\/latex] where [latex]\\begin{align}x={x}^{\\prime }\\cos \\theta -{y}^{\\prime }\\sin \\theta \\end{align}[\/latex] and [latex]\\begin{align}y={x}^{\\prime }\\sin \\theta +{y}^{\\prime }\\cos \\theta \\end{align}[\/latex].<\/li>\n
    2. Substitute the expression for [latex]x[\/latex] and [latex]y[\/latex] into in the given equation, then simplify.<\/li>\n
    3. Write the equations with [latex]\\begin{align}{x}^{\\prime }\\end{align}[\/latex] and [latex]\\begin{align}{y}^{\\prime }\\end{align}[\/latex] in standard form.<\/li>\n<\/ol>\n<\/section>\n
      Find a new representation of the equation [latex]2{x}^{2}-xy+2{y}^{2}-30=0[\/latex] after rotating through an angle of [latex]\\theta =45^\\circ[\/latex].<\/p>\n