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 \(O A B\).

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

1. Find the exact value of the cosine of angle \(B A C\).
2. Hence find the exact value of the area of triangle \(A B C\).
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 \(a x + b y + c z = d\).

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.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of triangle \(P Q R\).
Given that

• \(\overrightarrow { P Q } = 2 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }\)
• \(\overrightarrow { P R } = 8 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k }\)
  1. Find \(\overrightarrow { R Q }\)
  2. Find the size of angle \(P Q R\), in degrees, to three significant figures.

7．Relative to a fixed origin \(O\) the points \(A , B\) and \(C\) have position vectors $$\mathbf { a } = - \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { and } \mathbf { c } = 6 \mathbf { i } + \frac { 16 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { respectively. }$$ （a）Find the cosine of angle \(A B C\) ． The quadrilateral \(A B C D\) is a kite \(K\) ．
（b）Find the area of \(K\) ． A circle is drawn inside \(K\) so that it touches each of the 4 sides of \(K\) ．
（c）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．
（d）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.