Question

# The vector →a×(→b×→c) isPerpendicular to →b and parallel to →cCoplanar with →b and →c and orthogonal to →a(c) Perpendicular to both →b and →cNone of the above

Solution

## The correct option is B Coplanar with →b and →c and orthogonal to →aLook 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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