I'm just kinda confused about a problem in my linear algebra textbook. Maybe one of you geniuses on here can help me out.
There are two vectors in the plane that have a y-coordinate of 2 and form a 20 degree angle with the vector {1, 2}. Find them both and plot them.
I attempted at using the formula for finding the angle between two vectors and reversing it to solve the unknown but I got a really weird number
If anyone could just point me in the right direction that would be great. Thanks.
Let $\;(x,2)\;$ be such a vector, then as $\;20^\circ=\frac\pi9\;\text{radians}\;$ , we get
$$\cos \frac\pi9=\frac{(x,2)\cdot(1,2)}{||(x,2)||\;||(1,2)||}=\frac{x+4}{\sqrt{x^2+4}\,\cdot\sqrt5}$$
Putting $\;\alpha:=\cos\frac\pi9\;$ and squaring, we get the quadratic:
$$5\alpha^2(x^2+4)=x^2+8x+16$$
Solve this now and find your $\;x$'s .