I am asked to find the angle $\alpha$ on this particular setup:
The equation of the line is
and point $A = (2,1,4)$.
Is that really possible to find out? Maybe there is some data missing?
Thank you.
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3If the hexagon is indeed regular, then it's full of six equilateral triangles. Line $r$ is cutting two of those triangles exactly in half. Angle $\alpha$ is, thus, half the angle in an equilateral triangle. So ... – Blue Dec 24 '15 at 19:14
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Yes, just follow what Blue said and the answer will be $30^\circ$. – SchrodingersCat Dec 24 '15 at 19:15
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@Blue yes I agree with you, and that approach is perfect to me. But is there a way to find that 30 degree angle doing some calculation? – bru1987 Dec 24 '15 at 19:35
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@bru1987: Since you're using vectors, you could use dot products to calculate $$\cos \alpha = \frac{\overrightarrow{FA}\cdot\overrightarrow{FB}}{|\overrightarrow{FA}| |\overrightarrow{FB}|}$$ (The calculation should give $\sqrt{3}/2$, so that you know $\alpha = 30^\circ$.) We don't seem to know what $F$ is, though. Well, the equation for $r$ gives us a direction vector $(1,-1,2)$ that we can use in place of $\overrightarrow{FB}$. That still leaves $\overrightarrow{FA}$. Perhaps $F$ is the point on $r$ corresponding to $\lambda = 0$? – Blue Dec 24 '15 at 19:42
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@Blue I see what you mean, and I asked myself those exact same questions. Maybe in this exercise we are supposed to find $\alpha$ using basic geometry and saying that it is 30 degrees. – bru1987 Dec 24 '15 at 19:48
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@Blue thank you for your answer! – bru1987 Dec 24 '15 at 19:48