Three vectors `vecA,vecB` and `vecC` add up to zero.Find which is false.

Options

A. `(vecAxxvecB)xxvecC` is not zero unless `vecB, vecC` are parallel
B. `(vecAxxvecB). vecC` is not zero unless `vecB, vecC` are parallel
C. If `vecA, vecB, vecC` define a plane , `(vecAxxvecB)xxvecC` is in that plane
D. `(vecAxxvecB).vecC =|vecA||vecB||vecC|` if `C^(2)=A^(2)+B^(2)`

Answer

B

Explanation

(b): Since `vecA+vecB+vecC=0`
which means that `vecA , vecB and vecC` are coplanar . Option (a): `(vecAxxvecB)xxvecC=vecA.(vecBxxvecC)`
If `vecB and vecC` are parallel, then `vecA.(vecBxxvecC)=0` or `vecAxx(vecBxxvecC)=vec0 and (vecAxxvecB)xxvecCne o` if `vecB and vecC` are not parallel . Option (a) is correct option (b): For three coplanar vectors,
`(vecAxxvecB)/vecC=(vecBxxvecC).vecA` if B | | C, then `vecB xxvecC=0`
`therefore (vecBxxvecC).vecA=0 rArr ` option (b) is false.
Option (c): Since `vecA+vecB+vecC=0` , so they from a triangle as show in the figure and hence `vecA, vecB, vecC` define a plane `(vecAxxvecB)xxvec(C)`.

Option (d):
For three coplanar vectors, `(vecA.xxvecB).vecC=0` always
If `C^(2)=A^(2)+B^(2)`
And `vecA+vecB+vecc=0` already
This implies that A and B are mutually perpendicular to each other .
`(vecAxxvecB).vecC=(AB sin 90^(@) hatP). vecC`
`=ABC cos 90^(@)=0`
(Angle between unit vector `hatP and hatC` is also `90^(@)`).
Option (d) is not correct.