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

Question

Let $\mathbf{u}, \mathbf{v}$ and $\mathbf{w}$ be three vectors in $R^3$. Then, any vector $Z \in \mathbf{R}^3$ can be written as $z=a \mathbf{u}+b \mathbf{v}+c \mathbf{w}$ for some scalars $a, b$ and $c$ if and only if, Each pair of $\mathbf{u}, \mathbf{v}$ and $\mathbf{w}$ are not parallel, Each of $\mathbf{u}, \mathbf{v}$ and $\mathbf{w}$ can be written as a linear combination of the other two, All have different magnitude and directions, None of the options are correct

MARKS App · Accepted Answer

As given vector $u, v$ and $\mathbf{w}$ given may be not parallel but they may be antiparallel So, $\mathbf{z} \neq a \mathbf{u}+b \mathbf{v}+c \mathbf{w}$ So first is incorrect. Also, if $\mathbf{u}=\mathbf{v}+\mathbf{w}$ $\begin{gathered} \mathbf{v}=\mathbf{u}+\mathbf{w} \ \mathbf{w}=\mathbf{u}+\mathbf{v} \end{gathered}$ Then, $\mathbf{u}+\mathbf{v}+\mathbf{w}=2(\mathbf{u}+\mathbf{v}+\mathbf{w})$ $\Rightarrow \quad \mathbf{u}+\mathbf{v}+\mathbf{w}=\mathbf{0} \neq \mathbf{z}$ So, option (b) is incorrect. Similarly, option (c) is incorrect.

Search any question & find its solution

Looking for more such questions to practice?