The diagonals of a parallelogram are given by $A=3i-4j-k$ and $B=2i+3j-6k$. Show that the parallelogram is a rhombus and determine the length of its side and measure of its angle.
2026-03-31 18:54:43.1774983283
Show that the parallelogram is a rhombus.
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$(3i-4j-k)(2i+3j-6k)=0$. Thus, our parallelogram is rhombus.
The length-side is $$\frac{1}{2}\sqrt{3^2+(-4)^2+(-1)^2+2^2+3^2+(-6)^2}=\frac{5}{2}\sqrt3.$$ Measures of different angles are $2\arctan\frac{\sqrt{3^2+(-4)^2+(-1)^2}}{\sqrt{2^2+3^2+(-6)^2}}=2\arctan\frac{\sqrt{26}}{7}$ and $2\arctan\frac{7}{\sqrt{26}}$.