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

\(\mathbf{a}=\begin{pmatrix}2\\ -1\\ 1\end{pmatrix},\quad \mathbf{b}=\begin{pmatrix}1\\ 1\\ 1\end{pmatrix},\quad \mathbf{c}=\begin{pmatrix}0\\ 1\\ -1\end{pmatrix},\quad \mathbf{d}=\begin{pmatrix}3\\ -2\\ 0\end{pmatrix}.\)

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

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

The points \(A, B\) and \(C\) have position vectors \(\mathbf{i}+2 \mathbf{j}+2 \mathbf{k}, 2 \mathbf{i}+4 \mathbf{j}+5 \mathbf{k}\) and \(2 \mathbf{i}+3 \mathbf{j}+4 \mathbf{k}\) respectively. Find \(\overrightarrow{A B} \times \overrightarrow{A C}\).

Deduce, in either order, the exact value of
(i) the area of the triangle \(A B C\),
(ii) the perpendicular distance from \(C\) to \(A B\).

The position vectors of points \(A, B, C\), relative to the origin \(O\), are \(\mathbf{a}, \mathbf{b}, \mathbf{c}\), where
\(\mathbf{a}=3 \mathbf{i}+2 \mathbf{j}-\mathbf{k}, \quad \mathbf{b}=4 \mathbf{i}-3 \mathbf{j}+2 \mathbf{k}, \quad \mathbf{c}=3 \mathbf{i}-\mathbf{j}-\mathbf{k} .\)

Find \(\mathbf{a} \times \mathbf{b}\) and deduce the area of the triangle \(O A B\).

Hence find the volume of the tetrahedron \(O A B C\), given that the volume of a tetrahedron is \(\frac{1}{3} \times\) area of base × perpendicular height.