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

Options

A. `(AxxB)xxC` is not zero unless B, C are parallel
B. `(AxxB).C` is not zero unless B, C are parallel
C. If A, B, C define a plane, `(AxxB)xxC` is in that plane
D. `(AxxB).C=|A||B||C| rarr C^(2)=A^(2)+B^(2)`

Answer

B, D

Explanation

Given A+B+C=0
Hence, we can say that A, B and C are in one plane and are represented by the three sides of a triangle taken in one order, Now consider the options one by one.
(a) We can write
`Bxx(A+B+C)=Bxx0=0`
`rArr " " BxxA+BxxB+BxxC=0`
`rArr " " BxxA+0+BxxC=0`
`rArr" "BxxA=-BxxC`
`rArr " " AxxB=BxxC`
`:. " " (AxxB)xxC=(BxxC)xxC`
It cannot be zero.
If B||C, then `BxxC=0`, than `(BxxC)xxC=0`.
(b) `(AxxB).C=(BxxAC).C=0` whatever be the positions of A, B and C. If B||C, then `BxxC=0`, then `(BxxC)xxC=0`.
(c) `(AxxB)=X=AB sin theta X`. The direction of X is perpendicular to the plane containing A and B. `(AxxB)xxC=XxxC`. Its direction is in the plane of A, B, and C.
(d) If `C^(2)=A^(2)+B^(2)`, then angle between A and B is `90^(@)`
`:. " " (AxxB).C`=(AB `sin 90^(@)` X).C=AB (X.C)
`" " ABC cos 90^(@)=0`