For any three non-zero vectors `veca,vecb` and `vec c` if `|(vecaxxvecb). vec c|=|veca||vecb||vec c|` then `veca.vecb+vecb.vec c+vecc .veca=`

Options

A. `vec(a).vec(b)=vec(b).vec(c)=vec(c).vec(a)!=0`
B. `vec(a).vec(b)=0` only
C. `vec(b).vec(c)=0` only
D. `vec(a).vec(b)=vec(b).vec(c)=vec(c).vec(a)=0`

Answer

D

Explanation

Refer to the figure.
`|(vec(a)xx vec(b)).vec(c)|` is triple dot product and is volume (V) of the parallelepiped whose adjacent edges are a, b, and c.
i.e., `V=|(vec(a)xx vec(b))|.vec(O)C`
`rArr V=|vec(a)||vec(b)|(sin theta)(cos phi)|vec(c)|`
where `0 le theta le pi` is the angle between `vec(a) and vec(b)`.
As given
`|(vec(a) vec(b)). vec(c)|=|vec(a)||vec(b)||vec(c)|`
`rArr | sin theta cos phi|. = 1`
`rArr sin theta = 1, cos phi = 1`
`rArr theta = (pi)/(2) , phi = 0`
`rArr vec(a). vec(b) . =0 = vec(b). vec(c) = vec(c). vec(a)`