8.4 Rotation of axes  (Page 2/8)

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 $xy$ term, which aligns the graphs with the x - and y -axes. When we add an $xy$ term, we are rotating the conic about the origin. If the x - and y -axes are rotated through an angle, say $\theta ,$ then every point on the plane may be thought of as having two representations: $\left(x,y\right)$ on the Cartesian plane with the original x -axis and y -axis, and $\left(x^{\prime },y^{\prime }\right)$ on the new plane defined by the new, rotated axes, called the x' -axis and y' -axis. See [link] .

We will find the relationships between $x$ and $y$ on the Cartesian plane with $x^{\prime }$ and $y^{\prime }$ on the new rotated plane. See [link] .

The original coordinate x - and y -axes have unit vectors $i$ and $j.$ The rotated coordinate axes have unit vectors $i^{\prime }$ and $j^{\prime }.$ The angle $\theta$ is known as the angle of rotation    . See [link] . We may write the new unit vectors in terms of the original ones.

Consider a vector $u$ in the new coordinate plane. It may be represented in terms of its coordinate axes.

Because $u=x^{\prime }{i}^{\prime }+y^{\prime }{j}^{\prime },$ we have representations of $x$ and $y$ in terms of the new coordinate system.