1. ## Vectors

Hello!

Q: With respect to an origin O, the points P and Q have position vectors p and q respectively, given by p = (cos t)i + (sin t)j - k, q = (cos 2t)i - (sin 2t)j + $\frac{1}{2}$k, where t is a real parameter such that 0 < t < $2\pi$. Given that p $\cdot$q = cos 3t - $\frac{1}{2}$, hence or otherwise, find the greatest value of angle POQ.

Thank you for helping!

2. The dot product of two vectors is given by
$|\vec{u}||\vec{v}| cos(\theta)$ where $/theta$ is the angle between the vectors.
|p|= $\sqrt{cos^2(t)+ sin^2(t)+ 1}= \sqrt{2}$ while |q|= [tex]\sqrt{cos^2(2t)+ sin^2(2t)+ 1/4}= \sqrt{2}{2}.

Since p $\cdot$q= cos(3t)- 1, we must have
$cos(3t)-1 = cos(\theta)$ so the problem is simply "what is the maximum value of cos(3t)-1 for t between 0 and [tex]2\pi[/itex]" and then take the arccosine of that. Since cosine ranges between -1 and 1, cos(3t)-1 ranges between -2 and 0. -1, the smallest possible value of cosine, is included in that and so the largest possible angle between p and q is simply the largest possible angle between any two vectors, 180 degrees.

3. Thank you for helping!

May I know what did you evaluate |q| to be? I thought |q| = $\sqrt{5/4}$?