Answer to Question #246751 in Analytic Geometry for ejohn

Question

If C = [2.5 cm, 80°], i.e., the magnitude and direction of C are 2.5 cm and 80°, D = [3.5 cm,

120°], and E = D – 2C, then what is the direction of E (to the nearest degree)?

 

Answer 1

$$\vec c=(2.5\cos 80\degree, 2.5\sin 80\degree)$$

$$\vec d=(3.5\cos 120\degree, 3.5\sin 120\degree)$$

$$\vec e=\vec d-2\vec c=(3.5\cos 120\degree, 3.5\sin 120\degree)$$

$$-2(2.5\cos 80\degree, 2.5\sin 80\degree)$$

$$=(3.5\cos 120\degree-5\cos 80\degree, 3.5\sin 120\degree-5\sin 80\degree)$$

$$\tan \theta=\dfrac{3.5\sin 120\degree-5\sin 80\degree}{3.5\cos 120\degree-5\cos 80\degree}$$

$$\theta=\tan^{-1}(\dfrac{3.5\sin 120\degree-5\sin 80\degree}{3.5\cos 120\degree-5\cos 80\degree})$$

$$\approx36\degree$$