If a, b, c are three unit vectors such that a x b = c, b x c = a,

1.	If a, b, c are three unit vectors such that a x b = c, b x c = a, c x a = b. show that a, b, c, form an orthonormal right handed traid of unit vectors.
Answer» Given \(\vec a\times\vec b=\vec c,\) \(\vec b\times\vec c=\vec a\) and \(\vec c\times\vec a=\vec d\). Considering the first equation, \(\vec c\) is the cross product of the vectors \(\vec a\) and \(\vec b\). By the definition of the cross product of two vectors, we have \(\vec c\) perpendicular to both \(\vec a\) and \(\vec b.\). Similarly, considering the second equation, we have \(\vec a\) perpendicular to both \(\vec b\) and \(\vec c\). Once again, considering the third equation, we have \(\vec b\) perpendicular to both \(\vec c\) and \(\vec a\). From the above three statements, we can observe that the vectors \(\vec a,\,\vec b\) and \(\vec c\) are mutually perpendicular. It is also said that \(\vec a,\,\vec b\) and \(\vec c\) are three unit vectors. Thus, \(\vec a,\,\vec b,\,\vec c\) form an orthonormal right handed triad of unit vectors.

If a, b, c are three unit vectors such that a x b = c, b x c = a, c x a = b. show that a, b, c, form an orthonormal right handed traid of unit vectors.

Answer»

Given \(\vec a\times\vec b=\vec c,\) \(\vec b\times\vec c=\vec a\) and \(\vec c\times\vec a=\vec d\).

Considering the first equation, \(\vec c\) is the cross product of the vectors \(\vec a\) and \(\vec b\).

By the definition of the cross product of two vectors, we have \(\vec c\) perpendicular to both \(\vec a\) and \(\vec b.\).

Similarly, considering the second equation, we have \(\vec a\) perpendicular to both \(\vec b\) and \(\vec c\).

Once again, considering the third equation, we have \(\vec b\) perpendicular to both \(\vec c\) and \(\vec a\).

From the above three statements, we can observe that the vectors \(\vec a,\,\vec b\) and \(\vec c\) are mutually perpendicular.

It is also said that \(\vec a,\,\vec b\) and \(\vec c\) are three unit vectors.

Thus, \(\vec a,\,\vec b,\,\vec c\) form an orthonormal right handed triad of unit vectors.