Question

Let $u,v$ and $w$ be three vectors in $R^3$. Then any vector $Z\in R^3$ can be written as $z=au+bv+cw$ for some scalars $a,b$ and $c$ if and only if

• A. Each pair of $u,v$ and $w$ are not parallel
• B. Each of $u,v$ and $w$ can be written as a linear combination of the other two
• C. All have different magnitude and directions
• D. $N$ one of the options are correct

Answer 1

Answer: (D)

As given vector $u,v$ and $w$ given may be not parallel but they may be antiparallel
So, $z\ne au+bv+cw$
So first is incorrect.
Also, if $u=v+w$
$v=u+w$
$w=u+v$
Then, $u+v+w=2(u+v+w)$
$\Rightarrow u+v+w=0\ne z$
So, option (b) is incorrect.
Similarly, option (c) is incorrect.