If the diagonals of a parallelogram are given by $3i+j-2k$ and $i-3j+4k$, then find the lengths of the sides of parallelogram.
I am proceeding by finding the angle between diagonals and length of diagonals and then using cosine rule of triangle to find length of sides. Is this approach correct? Is there any other method?
Your way is correct , but I think the following way a bit of better.
One of lengths-sides is $$\frac{1}{2}|3i+j-2k-(i-3j+4k)|=\frac{1}{2}|2i+4j-6k|=$$ $$=|i+2j-3k|=\sqrt{1^2+2^2+(-3)^2}=\sqrt{14}$$ and the second it's $$\frac{1}{2}|3i+j-2k+(i-3j+4k)|=\frac{1}{2}|4i-2j+2k|=$$ $$=|2i-j+k|=\sqrt{6}.$$ Done!
Note that if you add the diagonals you will get twice one of the side. To see why just draw the parallelogram and stack another one on top of it. To find the length of the other side stack them next to each other and add the negative of one diagonal to the other. Once you find the vectors for the sides (be careful twice the side) you should be able to find it's length.
