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?
Best Answer
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!