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)
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2 thoughts on “Vectors and planes

  1. Pingback: Constrained Inclined plane « Beyond the Standard Model Pub

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