Exercise 3.7.4

Hint: Problem 7.4 and Problem 7.5, consider $\det <!--\; FUNCTION\; APPLICATION\; -->\mathit{RA}=(\det <!--\; FUNCTION\; APPLICATION\; -->R)(\det <!--\; FUNCTION\; APPLICATION\; -->A)=\det <!--\; FUNCTION\; APPLICATION\; -->A$, where $R$ is the rotation matrix with its determinant equal to 1. For proof of the parallelogram area, we can also use parameter angle, i.e., $v_1={[x_1,y_1]}^T={[v_1\cos <!--\; FUNCTION\; APPLICATION\; -->\alpha ,v_1\sin <!--\; FUNCTION\; APPLICATION\; -->\alpha ]}^T$, $v_2={[x_2,y_2]}^T={[v_2\cos <!--\; FUNCTION\; APPLICATION\; -->\beta ,v_2\sin <!--\; FUNCTION\; APPLICATION\; -->\beta ]}^T$. $v_1,v_2$ are the lengths of $v_1,v_2$, respectively. $\alpha ,\beta$ represents the angle between the vector and $x$-axis positive direction. Then

$\beta -\alpha$ is the angle from $v_1$ to $v_2$.