Rotation of Axes: Learn It 2

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Rotation of Axes: Learn It 2

Finding a New Representation of the Given Equation after Rotating through a Given Angle

Until now, we have looked at equations of conic sections without an [latex]xy[/latex] term, which aligns the graphs with the x– and y-axes. When we add an [latex]xy[/latex] term, we are rotating the conic about the origin. If the x– and y-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-axis and y-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’-axis and y’-axis.

The graph of the rotated ellipse [latex]{x}^{2}+{y}^{2}-xy - 15=0[/latex]

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.

The Cartesian plane with x- and y-axes and the resulting x′− and y′−axes formed by a rotation by an angle [latex]\theta [/latex].

The original coordinate x– and y-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. We may write the new unit vectors in terms of the original ones.

[latex]\begin{align}&{i}^{\prime }=i\cos \theta +j\sin \theta \\ &{j}^{\prime }=-i\sin \theta +j\cos \theta \end{align}[/latex]

Relationship between the old and new coordinate planes.

Consider a vector [latex]u[/latex] in the new coordinate plane. It may be represented in terms of its coordinate axes.

[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]

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.

[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]

equations of rotation

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-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]

[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]

How To: Given the equation of a conic, find a new representation after rotating through an angle.

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].
Substitute the expression for [latex]x[/latex] and [latex]y[/latex] into in the given equation, then simplify.
Write the equations with [latex]\begin{align}{x}^{\prime }\end{align}[/latex] and [latex]\begin{align}{y}^{\prime }\end{align}[/latex] in standard form.

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].

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].

Because [latex]\theta =45^\circ[/latex],

[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]

and

[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]

Substitute [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].

[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]

Simplify.

[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]

Write the equations with [latex]\begin{align}{x}^{\prime }\end{align}[/latex] and [latex]\begin{align}{y}^{\prime }\end{align}[/latex] in the standard form.

[latex]\begin{align}\frac{{{x}^{\prime }}^{2}}{20}+\frac{{{y}^{\prime }}^{2}}{12}=1\end{align}[/latex]

This equation is an ellipse. Figure 6 shows the graph.