Let \(\mathbf{u}, \mathbf{v}\) and \(\mathbf{w}\) be three vectors in \(R^3\). Then, any vector \(Z \in…
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
Solution
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.