Question

A body is in equilibrium under the action of three force vectors $\overrightarrow{A},\overrightarrow{B}$ and $\overrightarrow{C}$ simultaneously Show that $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{C}=\overrightarrow{C}\times \overrightarrow{A}$

Answer 1

An algebraic argument$:$ start with the given information$:-$ $A+B+C=0.$

cross with one of the vectors$,$ say$A.$ Note that $A\times A=0.$

Note that $B\times A=-A\times B.$

conclude that $A\times B=C\times A.$

Similarly for the other equations$.$

A geometric argument$:$ The three vectors are the sides of a triangle in order$.$ The area of the triangle is half the cross product of two of them in that order$.$