Question

A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u – and v -axes.R is the parallelogram with vertices \((0,0),(4,3),(2,4),( - 2,1)\)

Short answer

The required equations are

\(x = \frac{{u + 2v}}{5}\) and \(y = \frac{{3v - u}}{{10}}\)

Step-by-step solution

Given data

We have given,

The parallelogram with vertices \({(0,0),(4,3),(2,4),( - 2,1)}\)

Draw the graph and Find sides of parallelogram

We have to find the equation of sides of the parallelogram,

Draw the rectangular uv plane

We can define the parallelogram as

\( - 10 < 3x - 4y < 0\;\;{\rm{And}}\;\;0 < x + 2y < 10\)

If we choose3x-4y=u and

x+2y=v, we will get

\( - 10 < u < 0\;\;{\rm{And}}\;\;0 < v < 10\)

Which is a rectangle in uv plane,

Final calculation

Let
\(3x - 4y = u \to \) Equation1
and

\(x + 2y = v \to \;\) Equation2

Multiply Equation \(2\) by \(2\) and add to Equation 1

\(\begin{aligned}3x - 4y + 2x + 4y = u + 2v\\5x = u + 2v\\x = \frac{{u + 2v}}{5}\end{aligned}\)

Multiply Equation \(\;2\) by \( - 3\) and add to Equation1

\(\begin{aligned}{3x - 4y - 3x - 6y = u - 3v}\\{ - 10y = u - 3v}\\{y = \frac{{3v - u}}{{10}}}\end{aligned}\)

Hence, the required equations are

\(x = \frac{{u + 2v}}{5}\) and \(y = \frac{{3v - u}}{{10}}\)