I am not getting any start , can anybody please help me in this .
Prove the vectors are vertices of a cube
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vector-spaces
1 Answers
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A matrix $Q$ is said to be orthogonal if $Q^TQ =I $ where $I $ is the identity matrix.
Now if your given matrix is orthogonal we have that $$\sum a_i ^2 =1, \sum b_i^2 =0, \sum c_I^2 =1$$ and that $$\sum a_ib_i = \sum b_ic_i = \sum a_ic_i =0$$ Thus we have that $\vec a , \vec b, \vec c $ are unit vectors.
If $P $ and $Q $ are the angles between $\vec a $ and $\vec b $ and between $\vec a $ and $\vec c $, then as $$\vec a\cdot \vec b = 0 = \vec a\cdot \vec c $$ $$\Rightarrow |\vec a||\vec b |\cos P =0 = |\vec a||\vec c|\cos Q $$ giving us $P=Q=\frac {\pi}{2} $. Similarly we can prove that $\vec b $ is perpendicular to $\vec c $.
Can you take it from here? Hope it helps.