1 The vectors $\mathbf { a } , \mathbf { b } , \mathbf { c }$ and $\mathbf { d }$ in $\mathbb { R } ^ { 3 }$ are given by

$$\mathbf { a } = \left( \begin{array} { l } 
1 \\
2 \\
1
\end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { l } 
2 \\
9 \\
0
\end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 
3 \\
3 \\
4
\end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 
0 \\
- 8 \\
3
\end{array} \right) .$$

(i) Show that $\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}$ is a basis for $\mathbb { R } ^ { 3 }$.\\

(ii) Express $\mathbf { d }$ in terms of $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$.\\

\hfill \mbox{\textit{CAIE FP1 2018 Q1}}