Question

If $A,B$and $C$ are three non-coplanar vectors, then $(A+B+C).\left[\right(A+B)\times (A+C\left)\right]$ equals?

• A. $0$
• B. $[A,B,C]$
• C. $2[A,B,C]$
• D. $-[A,B,C]$

Answer 1

The correct option is D

$-[A,B,C]$

Find the value of $\mathbf{(}\mathit{A}\mathbf{+}\mathit{B}\mathbf{+}\mathit{C}\mathbf{)}\mathbf{.}\mathbf{\left[}\mathbf{\right(}\mathit{A}\mathbf{+}\mathit{B}\mathbf{)}\mathbf{\times }\mathbf{(}\mathit{A}\mathbf{+}\mathit{C}\mathbf{\left)}\mathbf{\right]}$:

Given that $A,B$and $C$ are three non-coplanar vector.

Now, $(A+B+C).\left[\right(A+B)\times (A+C\left)\right]$

$$\begin{gathered} =(A+B+C).(B\times A+A\times C+B\times C)\mathbf{}\mathbf{\left[}\mathit{E}\mathit{x}\mathit{p}\mathit{a}\mathit{n}\mathit{s}\mathit{i}\mathit{o}\mathit{n}\mathbf{\right]}\mathbf{} \\ =(A+B+C).(B\times A)+(A+B+C).(A\times C)+(A+B+C).(B\times C) \\ =C.(B\times A)+B.(A\times C)+A.(B\times C) \\ =-[A,B,C]-[A,B,C]+[A,B,C] \\ =-[A,B,C] \end{gathered}$$

Hence, the correct option is D.