Area of triangle or parallelogram using vectors

6 \includegraphics[max width=\textwidth, alt={}, center]{8c358a10-a3e1-47b5-ae62-30ba6b76c167-3_655_1011_255_566} The diagram shows triangle \(A B C\) where \(A B = 5 \mathrm {~cm} , A C = 4 \mathrm {~cm}\) and \(B C = 3 \mathrm {~cm}\). Three circles with centres at \(A , B\) and \(C\) have radii \(3 \mathrm {~cm} , 2 \mathrm {~cm}\) and 1 cm respectively. The circles touch each other at points \(E , F\) and \(G\), lying on \(A B , A C\) and \(B C\) respectively. Find the area of the shaded region \(E F G\).

7 \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-06_648_586_255_244} The diagram shows the parallelogram \(O A C B\) where \(\overrightarrow { O A } = 2 \mathbf { i } + 4 \mathbf { j }\) and \(\overrightarrow { O B } = 4 \mathbf { i } - 3 \mathbf { j }\).

1. Show that \(\cos A O B = - \frac { 2 \sqrt { 5 } } { 25 }\).
2. Hence find the exact value of \(\sin A O B\).
3. Determine the area of \(O A C B\).

\includegraphics{figure_2} The diagram shows a triangular pyramid \(ABCD\). It is given that $$\overrightarrow{AB} = 3\mathbf{i} + \mathbf{j} + \mathbf{k}, \quad \overrightarrow{AC} = \mathbf{i} - 2\mathbf{j} - \mathbf{k} \quad \text{and} \quad \overrightarrow{AD} = \mathbf{i} + 4\mathbf{j} - 7\mathbf{k}.$$

1. Verify, showing all necessary working, that each of the angles \(DAB\), \(DAC\) and \(CAB\) is \(90°\). [3]
2. Find the exact value of the area of the triangle \(ABC\), and hence find the exact value of the volume of the pyramid. [4]

[The volume \(V\) of a pyramid of base area \(A\) and vertical height \(h\) is given by \(V = \frac{1}{3}Ah\).]