I have three vectors $u_1,u_2,u_3$,

$$ ||u_1||=3\\ ||u_2||=1\\ ||u_3||=2\\ \angle u_1,u_2=\frac{\pi}{3}\\ \angle u_3,u_1=\frac{\pi}{4}\\ \angle u_3,u_2=\frac{\pi}{4}. $$

I am asked to calculate the length of $u_1+u_2+u_3$ and the area of the parallellogram spanned by $u_1+u_2$ and $u_3$.

My problem is with calculating the length of the vector sums. I don't believe I can use the normal pythagorean theorem since the vectors aren't all orthogonal to each other, but I assume I could use some generalized version of the Pythagorean theorem. However, since we haven't really learned that in the course I am taking, I don't think that's the point. Is there another way to do this using the most elementary concepts of linear algebra such as dot products, vector products etc.? I have a feeling the answer is really obvious but I just don't see it.


By $ \cdot $ I denote the usual scalar product in $3D$.


  1. We have $u_i \cdot u_j =||u_i||||u_j|| \cos \angle u_i,u_j$.

  2. $||u_1+u_2+u_3||^2=(u_1+u_2+u_3) \cdot (u_1+u_2+u_3) $.

Can you proceed ?

  • $\begingroup$ Ah, of course! I guess it was just as obvious as I imagined it would be. Thanks a lot. $\endgroup$ – Chisq Aug 16 '18 at 10:28

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