Practicing Success
Statement-1: Let $\vec a, \vec b, \vec c$ be three coterminous edges of a parallelopiped of volume 2 cubic units and r is any vector in space, then $|(\vec r.\vec a) (\vec b×\vec c)+(\vec r.\vec b) (\vec c×\vec a)+(\vec r. \vec c) (\vec a×\vec b)=2|\vec r|$ Statement-2: Any vector in space can be written as a linear combination of three non-coplanar vectors. |
Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. Statement-1 is True, Statement-2 is True; Statement-2 is not a correct explanation for Statement-1. Statement-1 is True, Statement-2 is False. Statement-1 is False, Statement-2 is True. |
Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. |
Clearly, statement-2 is true. $\begin{bmatrix}\vec a×\vec b&\vec b×\vec c&\vec c×\vec a\end{bmatrix}=[\vec a\,\,\vec b\,\,\vec c]^2 = 4≠0$ $⇒\vec a×\vec b, \vec b×\vec c, \vec c×\vec a$ are non-coplanar. Using statement-2, we have $\vec r = x(\vec a×\vec b) + y (\vec b×\vec c) +z (\vec c×\vec a)$ ...(i) Taking dot products with $\vec a, \vec b$ and $\vec c$ respectively, we get $\vec r.\vec a=y[\vec a\,\,\vec b\,\,\vec c], \vec r.\vec b =z[\vec a\,\,\vec b\,\,\vec c]$ and $\vec r.\vec c =[\vec a\,\,\vec b\,\,\vec c]$ Substituting the values of x, y, z in (i), we get $\vec r[\vec a\,\,\vec b\,\,\vec c]=(\vec r.\vec a) (\vec b×\vec c) + (\vec r.\vec b) (\vec c×\vec a) + (\vec r.\vec c) (\vec a×\vec b)$ $⇒\left|(\vec r.\vec a) (\vec b×\vec c) + (\vec r.\vec b) (\vec c×\vec a) + (\vec r.\vec c) (\vec a×\vec b)\right|$ $⇒|\vec r||\vec a\,\,\vec b\,\,\vec c|$ $⇒\left|(\vec r.\vec a) (\vec b×\vec c) + (\vec r.\vec b) (\vec c×\vec a) + (\vec r.\vec c) (\vec a×\vec b)\right|=2|\vec r|$ |