Area calculations using vectors

5. Relative to a fixed origin \(O\), the point \(A\) has position vector \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\), the point \(B\) has position vector \(5 \mathbf { i } + \mathbf { j } + \mathbf { k }\), and the point \(C\) has position vector \(7 \mathbf { i } - \mathbf { j }\).

1. Find the cosine of angle \(A B C\).
2. Find the exact value of the area of triangle \(A B C\). The point \(D\) has position vector \(7 \mathbf { i } + 3 \mathbf { k }\).
3. Show that \(A C\) is perpendicular to \(C D\).
4. Find the ratio \(A D : D B\).

6. Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { c } 6 \\ 3 \\ - 6 \end{array} \right)\) respectively. Find, in exact, simplified form,

1. the cosine of \(\angle A O B\),
2. the area of triangle \(O A B\),
3. the shortest distance from \(A\) to the line \(O B\).
   6. continued

3 A triangle ABC has vertices \(\mathrm { A } ( - 2,4,1 ) , \mathrm { B } ( 2,3,4 )\) and \(\mathrm { C } ( 4,8,3 )\). By calculating a suitable scalar product, show that angle ABC is a right angle. Hence calculate the area of the triangle.