# Vectors and planes

Date of the lesson: 31 Oct 2011

In order to solve exercises involving vectors, trigonometric identities come very handy. For instance:

$\cos(\pi/2-\theta)=\sin(\theta)$

$\sin(\pi/2-\theta)=\cos(\theta)$

And don’t forget the prosthaphaeresis formulae!

One important relationship is the Law of Sines:

$\frac{\sin\alpha}{a}=\frac{\sin\beta}{b}=\frac{\sin\gamma}{c}$

Summing vector is easy as long as you project each vector along the coordinate axes. Five steps north, two steps east, three steps south…

How to fly a plane from Bologna to Prague:

http://www.algebralab.org/Word/Word.aspx?file=Trigonometry_ResultantsDotProducts.xml

Wanna try to land your cessna airplane? Click this link for a nice flash game. Why don’t you code an android applet for you mobile phone instead?

### Exercises

1. Calculate $\vec{a}\cdot\vec{b}$ and $\vec{a}\times\vec{b}$, where: $\vec{a}=5\hat{\imath}-6\hat{\jmath}+3\hat{k}$, $\vec{b}=-2\hat{\imath}+7\hat{\jmath}+4\hat{k}$
2. The resultant vector $\vec{r}=\vec{u}+\vec{v}$ has length $r=100$. The angle between $\vec{u}$ and $\vec{v}$ is $\theta_{uv}=\pi/2$. Knowing that the angle $\alpha_{rv}=\pi/6$, find the length of $\vec{u}$ and $\vec{v}$.
3. Calculate the angle between $\vec{a}=2\hat{\imath}+2\hat{\jmath}-\hat{k}$, $\vec{b}=6\hat{\imath}-3\hat{\jmath}+2\hat{k}$
4. For the die-hard: calculate the area of the triangle defined by vertices A(-1,2,3) B(2,-1,1) C(1,3,2)