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The diagram shows a triangular pyramid \(A B C D\). It is given that
$$\overrightarrow { A B } = 3 \mathbf { i } + \mathbf { j } + \mathbf { k } , \quad \overrightarrow { A C } = \mathbf { i } - 2 \mathbf { j } - \mathbf { k } \quad \text { and } \quad \overrightarrow { A D } = \mathbf { i } + 4 \mathbf { j } - 7 \mathbf { k }$$
- Verify, showing all necessary working, that each of the angles \(D A B , D A C\) and \(C A B\) is \(90 ^ { \circ }\).
- Find the exact value of the area of the triangle \(A B C\), and hence find the exact value of the volume of the pyramid.
[0pt]
[The volume \(V\) of a pyramid of base area \(A\) and vertical height \(h\) is given by \(V = \frac { 1 } { 3 } A h\).]