Question

HELP PLS>>> The vector (3, - 7) ) is rotated by an angle of (3pi)/4 radians and then reflected across the y-axis. If the resulting vector is [a/b], then a=__ and b=__

Answer 1

Answer:

$a \approx -2.826$, $b \approx 7.072$

Step-by-step explanation:

First, the vector must be transformed into its polar form:

$r = \sqrt{3^{2}+ (-7)^{2}}$

$r \approx 7.616$

$\theta \approx 2\pi - \tan^{-1}\left(\frac{7}{3} \right)$

$\theta \approx 1.629\pi$

Let assume that vector is rotated counterclockwise. The new angle is:

$\theta' = \theta + \frac{3\pi}{4}$

$\theta' = 2.379\pi$

Which is coterminal with $\theta'' = 0.379\pi$. The reflection across y-axis is:

$\theta''' = \pi - \theta''$

$\theta''' = 0.621\pi$

The equivalent vector in rectangular coordinates is:

$a = 7.616\cdot \cos 0.621\pi$

$a \approx -2.826$

$b = 7.616\cdot \sin 0.621\pi$

$b \approx 7.072$