Angle between vectors using scalar product

6 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 7 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k }$$

1. Use a scalar product to find angle \(O A B\).
2. Find the area of triangle \(O A B\).

6 Relative to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 3 \\ 1 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 5 \\ 3 \\ - 2 \end{array} \right)$$

1. Using a scalar product, find the cosine of angle \(B A C\).
2. Hence find the area of triangle \(A B C\). Give your answer in a simplified exact form.

7.Relative to a fixed origin \(O\) the points \(A , B\) and \(C\) have position vectors $$\mathbf { a } = - \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { and } \mathbf { c } = 6 \mathbf { i } + \frac { 16 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { respectively. }$$

1. Find the cosine of angle \(A B C\) . The quadrilateral \(A B C D\) is a kite \(K\) .
2. Find the area of \(K\) . A circle is drawn inside \(K\) so that it touches each of the 4 sides of \(K\) .
3. Find the radius of the circle,giving your answer in the form \(p \sqrt { } ( q ) - q \sqrt { } ( p )\) ,where \(p\) and \(q\) are positive integers.
4. Find the position vector of the point \(D\) .
   (Total 18 marks)

3

1. 1. Find \(\left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right) \times \left( \begin{array} { c } 3 \\ 5 \\ - 2 \end{array} \right)\).
   2. State a geometrical relationship between the answer to part (a)(i) and the vectors \(\left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { c } 3 \\ 5 \\ - 2 \end{array} \right)\).
   3. Verify the relationship stated in part (a)(ii).
2. Find the angle between the vectors \(2 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) and \(4 \mathbf { i } - \mathbf { j } + 8 \mathbf { k }\).

5

1. Given that \(\mathbf { u } = \left( \begin{array} { r } - 2 \\ 1 \\ 2 \end{array} \right) , \mathbf { v } = \left( \begin{array} { l } a \\ 0 \\ 1 \end{array} \right)\) and \(\mathbf { u } \times \mathbf { v } = \left( \begin{array} { l } 1 \\ b \\ 3 \end{array} \right)\), find \(a\) and \(b\).
2. Using \(\mathbf { u } \times \mathbf { v }\), determine the angle between the vectors \(\mathbf { u }\) and \(\mathbf { v }\), given that this angle is acute.

9 Three points \(A , B\) and \(C\) have coordinates \(( 1,0,7 ) , ( 13,9,1 )\) and \(( 2 , - 1 , - 7 )\) respectively.

1. Use a scalar product to find angle \(A C B\).
2. Hence find the area of triangle \(A C B\).
3. Show that a vector equation of the line \(A B\) is given by \(\mathbf { r } = \mathbf { i } + 7 \mathbf { k } + \lambda ( 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )\), where \(\lambda\) is a scalar parameter.

Referred to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors given by $$\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}, \quad \overrightarrow{OB} = 2\mathbf{i} + 4\mathbf{j} + \mathbf{k} \quad \text{and} \quad \overrightarrow{OC} = 3\mathbf{i} + 5\mathbf{j} - 3\mathbf{k}.$$

1. Find the exact value of the cosine of angle \(BAC\). [4]
2. Hence find the exact value of the area of triangle \(ABC\). [3]
3. Find the equation of the plane which is parallel to the \(y\)-axis and contains the line through \(B\) and \(C\). Give your answer in the form \(ax + by + cz = d\). [5]

\includegraphics{figure_1} Figure 1 shows a sketch of triangle \(PQR\). Given that • \(\overrightarrow{PQ} = 2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}\) • \(\overrightarrow{PR} = 8\mathbf{i} - 5\mathbf{j} + 3\mathbf{k}\)

1. Find \(\overrightarrow{RQ}\) [2]
2. Find the size of angle \(PQR\), in degrees, to three significant figures. [3]

\includegraphics{figure_5} Figure 1 shows a sketch of a triangle \(ABC\). Given \(\overrightarrow{AB} = 2\mathbf{i} + 3\mathbf{j} + \mathbf{k}\) and \(\overrightarrow{BC} = \mathbf{i} - 9\mathbf{j} + 3\mathbf{k}\), show that \(\angle BAC = 105.9°\) to one decimal place. [5]