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Question

The vector a×(b×c) is
  1. Perpendicular to b and parallel to c
  2. Coplanar with b and c and orthogonal to a
  3. (c) Perpendicular to both b and c
  4. None of the above


Solution

The correct option is B Coplanar with b and c and orthogonal to a
Look its simple.
Let u=a×(b×c)
­­­From the definition of cross product u  is perpendicular to a and b×c                              …. (1)
Now u is perpendicular to b×c.
 
See the figure. If b and c are the vectors (mentioned in blue) in the plane as shown then b×c will be perpendicular to that plane as shown by orange arrow. This orange arrow is in the direction of b×c
Now from statement 1 we can say that
u is perpendicular to the plane of b×c. The plane perpendicular to the plane of b×c (the orange arrow) is the plane which contains b and c
This implies u  lies in the plane of b and c.
Also from the statement (1), u is orthogonal to a                                   (Definition of cross product).
So u is coplanar with b and c and orthogonal to a.
 

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