Question

Let $u, v$ and $w$ be three vectors in $R^{3}$. Then any vector $Z \in R^{3}$ can be written as $z=a u+b v+c w$ 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, $z \neq a u +b v+ c w$
So first is incorrect.
Also, if $u=v+ w$
$v=u+ w$
$w=u+ v$
Then, $u+v+w=2(u+ v+ w)$
$\Rightarrow u+ v+ w=0 \neq z$
So, option (b) is incorrect.
Similarly, option (c) is incorrect.