Final answer:
The expression (a×b)×c= 1/3 |b| |c | a involves the cross product of vectors and the magnitude of vectors. No two vectors are collinear, so the cross product (a×b) is nonzero and perpendicular to both a and b. The angle between b and c is sin^(-1)(1/3), so sinθ equals 1/3. The correct answer is D. 1/3.
Step-by-step explanation:
The given expression, (a×b)×c= 1/3 |b| |c | a, involves the cross product of vectors and the magnitude of vectors.
The cross product of two vectors a and b is given by the formula a×b = |a||b|sinθ, where θ is the angle between the vectors.
In this case, (a×b)×c = 1/3 |b| |c | a. Since no two vectors are collinear, the cross product (a×b) is nonzero and perpendicular to both a and b.
Therefore, the expression becomes |(a×b)| |c| = 1/3 |b| |c| |a|.
Dividing both sides by |b| |c|, we get |a×b| = 1/3 |a|.
Since a and b are non-collinear vectors, the angle between them, which is also the angle between b and c, is θ = sin^(-1)(1/3).
Therefore, the value of sinθ is 1/3, which corresponds to option D.