Question

If `overlinea`, `overlineb` and `overlinec` are three non-coplanar vectors then `(overlinea + overlineb + overlinec).[(overlinea + overlineb) xx (overlinea + overlinec)]` = ______

Options

• 0

• `[overlinea   overlineb  overlinec]`

• `-[overlinea   overlineb  overlinec]`

• 2`[overlinea   overlineb  overlinec]`

Answer 1

`-[overlinea   overlineb  overlinec]`

Explanation:

`[overlinea + overlineb + overlinec].[(overlinea + overlineb) xx (overlinea + overlinec)]`

= `(overlinea + overlineb).[(overlinea + overlineb) xx (overlinea + overlinec)] + overlinec.[(overlinea + overlineb) xx (overlinea + overlinec)]`

= 0 + `[overlinec  overlinea + overlineb  overlinea + overlinec]`

= `[overlinec  overlinea  overlinea + overlinec] + [overlinec  overlineb  overlinea + overlinec]`

= `[overlinec  overlinea  overlinea] + [overlinec  overlinea  overlinec] + [overlinec  overlineb  overlinea] + [overlinec  overlineb  overlinec]`

= `0 + 0 + [overlinec  overlineb  overlinea] + 0 = -[overlinea  overlineb  overlinec]`