Relative to a fixed origin \(O\),
the point \(A\) has position vector \(( 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } )\),
the point \(B\) has position vector ( \(4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }\) ),
and the point \(C\) has position vector ( \(a \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k }\) ), where \(a\) is a constant and \(a < 0 D\) is the point such that \(\overrightarrow { A B } = \overrightarrow { B D }\).