यदि a + b + c = 0 तो सिद्ध करें कि a x b = b x c = c x a.

Question

यदि \(\vec{a}+\vec{b}+\vec{c}=0\) तो सिद्ध करें कि \(\vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{c} \times \vec{a}\).

Answer 1

If \(\vec{a}+\vec{b}+\vec{c}=0 \)

\(\Rightarrow \vec{a}+\vec{b}=-\vec{c}\)

\(\Rightarrow(\vec{a}+\vec{b}) \times \vec{b}=-\vec{c} \times \vec{b} \)

\(\Rightarrow(\vec{a} \times \vec{b})+(\vec{b} \times \vec{b})=-\vec{c} \times \vec{b} \)

\(\Rightarrow \vec{a} \times \vec{b}+0=-(\vec{c} \times \vec{b}) \)

\( \Rightarrow \vec{a} \times \vec{b}=\vec{b} \times \vec{c} \) .....(1)

Again, \(\vec{a}+\vec{b}+\vec{c}=0 \Rightarrow \vec{b}+\vec{c}=-\vec{a}\)

\(\Rightarrow(\vec{b}+\vec{c}) \times \vec{c}=-\vec{a} \times \vec{c} \)

\(\Rightarrow(\vec{b} \times \vec{c})+(\vec{c} \times \vec{c})=\vec{c} \times \vec{a}\)

\(\Rightarrow \vec{b} \times \vec{c}=\vec{c} \times \vec{a} ; \quad (\vec{c} \times \vec{c}=0)\) ....(2)

From (1) and (2), \(\vec{a} \times \vec{b}=\vec{b} \times \vec{c}=\vec{c} \times \vec{a}\)