Explanation
From the previous analysis, we know that a,b, and c are mutually orthogonal unit vectors. Therefore, ∣a∣=∣b∣=∣c∣=1 and a⋅b=b⋅c=c⋅a=0.
1. Evaluating Statement I:
Substitute the given identities a×b=c and b×c=a:
(a×b)⋅c=c⋅c=∣c∣2=1
(b×c)⋅a=a⋅a=∣a∣2=1
For the right side, since a,b,c form a right-handed system with a×b=c, it follows that c×a=b:
(c×a)⋅b=b⋅b=∣b∣2=1
Substituting these into the equation:
1+1=1⟹2=1
This is false. Therefore, Statement I is incorrect.
2. Evaluating Statement II:
Substitute the given identities into the expression:
{(a×b)×(b×c)}⋅b={c×a}⋅b
As established, c×a=b:
{b}⋅b=∣b∣2
Since b is a unit vector, ∣b∣2=1.
Therefore, 1=1, which is true. Statement II is correct.
Conclusion: Only Statement II is correct.
Correct Option: (b)
