Angles between vectors

6 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } + 8 \mathbf { k } , \quad \overrightarrow { O C } = - \mathbf { i } - 2 \mathbf { j } + 10 \mathbf { k }$$

1. Use a scalar product to find angle \(A B C\).
2. Find the perimeter of triangle \(A B C\), giving your answer correct to 2 decimal places.

8

1. Find the angle between the vectors \(3 \mathbf { i } - 4 \mathbf { k }\) and \(2 \mathbf { i } + 3 \mathbf { j } - 6 \mathbf { k }\). The vector \(\overrightarrow { O A }\) has a magnitude of 15 units and is in the same direction as the vector \(3 \mathbf { i } - 4 \mathbf { k }\). The vector \(\overrightarrow { O B }\) has a magnitude of 14 units and is in the same direction as the vector \(2 \mathbf { i } + 3 \mathbf { j } - 6 \mathbf { k }\).
2. Express \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Find the unit vector in the direction of \(\overrightarrow { A B }\).

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

1. Use a vector method to find angle \(A O B\). The point \(C\) is such that \(\overrightarrow { A B } = \overrightarrow { B C }\).
2. Find the unit vector in the direction of \(\overrightarrow { O C }\).
3. Show that triangle \(O A C\) is isosceles.

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

1. Find \(\overrightarrow { A C }\).
2. The point \(M\) is the mid-point of \(A C\). Find the unit vector in the direction of \(\overrightarrow { O M }\).
3. Evaluate \(\overrightarrow { A B } \cdot \overrightarrow { A C }\) and hence find angle \(B A C\).

7
\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-3_636_1047_1153_550} The diagram shows a triangular prism with a horizontal rectangular base \(A D F C\), where \(C F = 12\) units and \(D F = 6\) units. The vertical ends \(A B C\) and \(D E F\) are isosceles triangles with \(A B = B C = 5\) units. The mid-points of \(B E\) and \(D F\) are \(M\) and \(N\) respectively. The origin \(O\) is at the mid-point of \(A C\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O C , O N\) and \(O B\) respectively.

1. Find the length of \(O B\).
2. Express each of the vectors \(\overrightarrow { M C }\) and \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Evaluate \(\overrightarrow { M C } \cdot \overrightarrow { M N }\) and hence find angle \(C M N\), giving your answer correct to the nearest degree.

8 The points \(A\) and \(B\) have position vectors \(\mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k }\) and \(- 5 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\) respectively, relative to an origin \(O\).

1. Use a scalar product to calculate angle \(A O B\), giving your answer in radians correct to 3 significant figures.
2. The point \(C\) is such that \(\overrightarrow { A B } = 2 \overrightarrow { B C }\). Find the unit vector in the direction of \(\overrightarrow { O C }\).

5
\includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-2_741_533_1279_808} The diagram shows a pyramid \(O A B C\) with a horizontal base \(O A B\) where \(O A = 6 \mathrm {~cm} , O B = 8 \mathrm {~cm}\) and angle \(A O B = 90 ^ { \circ }\). The point \(C\) is vertically above \(O\) and \(O C = 10 \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O C\) as shown. Use a scalar product to find angle \(A C B\).

6 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\) and \(5 \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k }\) respectively.

1. Use a scalar product to find angle \(B O A\). The point \(C\) is the mid-point of \(A B\). The point \(D\) is such that \(\overrightarrow { O D } = 2 \overrightarrow { O B }\).
2. Find \(\overrightarrow { D C }\).

11 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 2 \mathbf { i } - \mathbf { j }\) and \(\overrightarrow { O B } = \mathbf { j } - 2 \mathbf { k }\).

1. Show that \(O A = O B\) and use a scalar product to calculate angle \(A O B\) in degrees.
   The midpoint of \(A B\) is \(M\). The point \(P\) on the line through \(O\) and \(M\) is such that \(P A : O A = \sqrt { 7 } : 1\).
2. Find the possible position vectors of \(P\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ 1 \\ - 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 0 \\ 4 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } - 3 \\ - 2 \\ 2 \end{array} \right)$$

1. The point \(D\) is such that \(A B C D\) is a trapezium with \(\overrightarrow { D C } = 3 \overrightarrow { A B }\). Find the position vector of \(D\).
2. The diagonals of the trapezium intersect at the point \(P\). Find the position vector of \(P\).
   \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-13_2725_35_99_20}
3. Using a scalar product, calculate angle \(A B C\).

5 The vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by $$\mathbf { a } = 3 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } \quad \text { and } \quad \mathbf { b } = 2 \mathbf { i } - \mathbf { j } - 5 \mathbf { k }$$ 5

1. Calculate a.b 5
2. \(\quad\) Calculate \(| \mathbf { a } |\) and \(| \mathbf { b } |\)
   \(| \mathbf { a } | =\) \(\_\_\_\_\)
   5
3. Calculate the acute angle between \(\mathbf { a }\) and \(\mathbf { b }\)
   Give your answer to the nearest degree.