1.10d Vector operations: addition and scalar multiplication

5 \includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-3_1070_754_255_699} The diagram shows a solid cylinder standing on a horizontal circular base, centre \(O\) and radius 4 units. The line \(B A\) is a diameter and the radius \(O C\) is at \(90 ^ { \circ }\) to \(O A\). Points \(O ^ { \prime } , A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\) lie on the upper surface of the cylinder such that \(O O ^ { \prime } , A A ^ { \prime } , B B ^ { \prime }\) and \(C C ^ { \prime }\) are all vertical and of length 12 units. The mid-point of \(B B ^ { \prime }\) is \(M\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O O ^ { \prime }\) respectively.

1. Express each of the vectors \(\overrightarrow { M O }\) and \(\overrightarrow { M C ^ { \prime } }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Hence find the angle \(O M C ^ { \prime }\).

8 The points \(A , B , C\) and \(D\) have position vectors \(3 \mathbf { i } + 2 \mathbf { k } , 2 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } , 2 \mathbf { j } + 7 \mathbf { k }\) and \(- 2 \mathbf { i } + 10 \mathbf { j } + 7 \mathbf { k }\) respectively.

1. Use a scalar product to show that \(B A\) and \(B C\) are perpendicular.
2. Show that \(B C\) and \(A D\) are parallel and find the ratio of the length of \(B C\) to the length of \(A D\).

9 Relative to an origin \(O\), the position vectors of the points \(A , B , C\) and \(D\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 1 \\ 3 \\ - 1 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ - 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { l } 4 \\ 2 \\ p \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } - 1 \\ 0 \\ q \end{array} \right) ,$$ where \(p\) and \(q\) are constants. Find

1. the unit vector in the direction of \(\overrightarrow { A B }\),
2. the value of \(p\) for which angle \(A O C = 90 ^ { \circ }\),
3. the values of \(q\) for which the length of \(\overrightarrow { A D }\) is 7 units.

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

1. Use a scalar product to find angle \(A O B\), correct to the nearest degree.
2. Find the unit vector in the direction of \(\overrightarrow { A B }\).
3. The point \(C\) is such that \(\overrightarrow { O C } = 6 \mathbf { j } + p \mathbf { k }\), where \(p\) is a constant. Given that the lengths of \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\) are equal, find the possible values of \(p\).

9 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 4 \\ 1 \\ - 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ 2 \\ - 4 \end{array} \right) .$$

1. Given that \(C\) is the point such that \(\overrightarrow { A C } = 2 \overrightarrow { A B }\), find the unit vector in the direction of \(\overrightarrow { O C }\). The position vector of the point \(D\) is given by \(\overrightarrow { O D } = \left( \begin{array} { l } 1 \\ 4 \\ k \end{array} \right)\), where \(k\) is a constant, and it is given that \(\overrightarrow { O D } = m \overrightarrow { O A } + n \overrightarrow { O B }\), where \(m\) and \(n\) are constants.
2. Find the values of \(m , n\) and \(k\).

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

1. Find the value of \(p\) for which \(O A\) and \(O B\) are perpendicular.
2. In the case where \(p = 6\), use a scalar product to find angle \(A O B\), correct to the nearest degree.
3. Express the vector \(\overrightarrow { A B }\) is terms of \(p\) and hence find the values of \(p\) for which the length of \(A B\) is 3.5 units.

6 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } - 8 \mathbf { j } + 4 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 7 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$$

1. Find the value of \(\overrightarrow { O A } \cdot \overrightarrow { O B }\) and hence state whether angle \(A O B\) is acute, obtuse or a right angle.
2. The point \(X\) is such that \(\overrightarrow { A X } = \frac { 2 } { 5 } \overrightarrow { A B }\). Find the unit vector in the direction of \(O X\).

10 \includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-4_552_629_842_758} The diagram shows the parallelogram \(O A B C\). Given that \(\overrightarrow { O A } = \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { O C } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k }\), find

1. the unit vector in the direction of \(\overrightarrow { O B }\),
2. the acute angle between the diagonals of the parallelogram,
3. the perimeter of the parallelogram, correct to 1 decimal place.

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.

4 \includegraphics[max width=\textwidth, alt={}, center]{53839c8c-07ea-4545-9c00-a6884aa2afc3-2_750_855_902_646} The diagram shows a prism \(A B C D P Q R S\) with a horizontal square base \(A P S D\) with sides of length 6 cm . The cross-section \(A B C D\) is a trapezium and is such that the vertical edges \(A B\) and \(D C\) are of lengths 5 cm and 2 cm respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(A D , A P\) and \(A B\) respectively.

1. Express each of the vectors \(\overrightarrow { C P }\) and \(\overrightarrow { C Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to calculate angle \(P C Q\).

6 Two vectors \(\mathbf { u }\) and \(\mathbf { v }\) are such that \(\mathbf { u } = \left( \begin{array} { c } p ^ { 2 } \\ - 2 \\ 6 \end{array} \right)\) and \(\mathbf { v } = \left( \begin{array} { c } 2 \\ p - 1 \\ 2 p + 1 \end{array} \right)\), where \(p\) is a constant.

1. Find the values of \(p\) for which \(\mathbf { u }\) is perpendicular to \(\mathbf { v }\).
2. For the case where \(p = 1\), find the angle between the directions of \(\mathbf { u }\) and \(\mathbf { v }\).

6 Relative to an origin \(O\), the position vectors of three points, \(A , B\) and \(C\), are given by $$\overrightarrow { O A } = \mathbf { i } + 2 p \mathbf { j } + q \mathbf { k } , \quad \overrightarrow { O B } = q \mathbf { j } - 2 p \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = - \left( 4 p ^ { 2 } + q ^ { 2 } \right) \mathbf { i } + 2 p \mathbf { j } + q \mathbf { k }$$ where \(p\) and \(q\) are constants.

1. Show that \(\overrightarrow { O A }\) is perpendicular to \(\overrightarrow { O C }\) for all non-zero values of \(p\) and \(q\).
2. Find the magnitude of \(\overrightarrow { C A }\) in terms of \(p\) and \(q\).
3. For the case where \(p = 3\) and \(q = 2\), find the unit vector parallel to \(\overrightarrow { B A }\).

6 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 3 \mathbf { i } + p \mathbf { j } + q \mathbf { k }$$ where \(p\) and \(q\) are constants.

1. State the values of \(p\) and \(q\) for which \(\overrightarrow { O A }\) is parallel to \(\overrightarrow { O B }\).
2. In the case where \(q = 2 p\), find the value of \(p\) for which angle \(B O A\) is \(90 ^ { \circ }\).
3. In the case where \(p = 1\) and \(q = 8\), find the unit vector in the direction of \(\overrightarrow { A B }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-3_465_554_255_794} The diagram shows three points \(A ( 2,14 ) , B ( 14,6 )\) and \(C ( 7,2 )\). The point \(X\) lies on \(A B\), and \(C X\) is perpendicular to \(A B\). Find, by calculation,

1. the coordinates of \(X\),
2. the ratio \(A X : X B\).

8 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-3_716_437_1137_854} The diagram shows a parallelogram \(O A B C\) in which $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ 3 \\ - 4 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right)$$

1. Use a scalar product to find angle \(B O C\).
2. Find a vector which has magnitude 35 and is parallel to the vector \(\overrightarrow { O C }\).

8 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { c } 3 p \\ 4 \\ p ^ { 2 } \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { c } - p \\ - 1 \\ p ^ { 2 } \end{array} \right)$$

1. Find the values of \(p\) for which angle \(A O B\) is \(90 ^ { \circ }\).
2. For the case where \(p = 3\), find the unit vector in the direction of \(\overrightarrow { B A }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-3_394_750_260_699} The diagram shows a trapezium \(A B C D\) in which \(B A\) is parallel to \(C D\). The position vectors of \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 3 \\ 4 \\ 0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 4 \\ 5 \\ 6 \end{array} \right)$$

1. Use a scalar product to show that \(A B\) is perpendicular to \(B C\).
2. Given that the length of \(C D\) is 12 units, find the position vector of \(D\).

7 The position vectors of points \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 6 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 2 \\ 4 \\ 7 \end{array} \right)$$

1. Show that angle \(B A C = \cos ^ { - 1 } \left( \frac { 1 } { 3 } \right)\).
2. Use the result in part (i) to find the exact value of the area of triangle \(A B C\).

10 Relative to an origin \(O\), the position vectors of points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ 1 \\ - 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 5 \\ - 1 \\ k \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2 \\ 6 \\ - 3 \end{array} \right)$$ respectively, where \(k\) is a constant.

1. Find the value of \(k\) in the case where angle \(A O B = 90 ^ { \circ }\).
2. Find the possible values of \(k\) for which the lengths of \(A B\) and \(O C\) are equal. The point \(D\) is such that \(\overrightarrow { O D }\) is in the same direction as \(\overrightarrow { O A }\) and has magnitude 9 units. The point \(E\) is such that \(\overrightarrow { O E }\) is in the same direction as \(\overrightarrow { O C }\) and has magnitude 14 units.
3. Find the magnitude of \(\overrightarrow { D E }\) in the form \(\sqrt { } n\) where \(n\) is an integer.

3 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } - 5 \mathbf { j } - 2 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 4 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }$$ The point \(C\) is such that \(\overrightarrow { A B } = \overrightarrow { B C }\). Find the unit vector in the direction of \(\overrightarrow { O C }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-12_775_823_260_662} The diagram shows a three-dimensional shape in which the base \(O A B C\) and the upper surface \(D E F G\) are identical horizontal squares. The parallelograms \(O A E D\) and \(C B F G\) both lie in vertical planes. The point \(M\) is the mid-point of \(A F\). Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O C\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards. The position vectors of \(A\) and \(D\) are given by \(\overrightarrow { O A } = 8 \mathbf { i }\) and \(\overrightarrow { O D } = 3 \mathbf { i } + 10 \mathbf { k }\).

1. Express each of the vectors \(\overrightarrow { A M }\) and \(\overrightarrow { G M }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to find angle \(G M A\) correct to the nearest degree.

8 The position vectors of points \(A\) and \(B\), relative to an origin \(O\), are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 6 \\ - 2 \\ - 6 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ k \\ - 3 \end{array} \right)$$ where \(k\) is a constant.

1. Find the value of \(k\) for which angle \(A O B\) is \(90 ^ { \circ }\).
2. Find the values of \(k\) for which the lengths of \(O A\) and \(O B\) are equal.
   The point \(C\) is such that \(\overrightarrow { A C } = 2 \overrightarrow { C B }\).
3. In the case where \(k = 4\), find the unit vector in the direction of \(\overrightarrow { O C }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-3_529_698_260_721} The diagram shows a pyramid \(O A B C\) with a horizontal triangular base \(O A B\) and vertical height \(O C\). Angles \(A O B , B O C\) and \(A O C\) are each right angles. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O C\) respectively, with \(O A = 4\) units, \(O B = 2.4\) units and \(O C = 3\) units. The point \(P\) on \(C A\) is such that \(C P = 3\) units.

1. Show that \(\overrightarrow { C P } = 2.4 \mathbf { i } - 1.8 \mathbf { k }\).
2. Express \(\overrightarrow { O P }\) and \(\overrightarrow { B P }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Use a scalar product to find angle \(B P C\).

7 Given that \(\mathbf { a } = \left( \begin{array} { r } 2 \\ - 2 \\ 1 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 2 \\ 6 \\ 3 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { c } p \\ p \\ p + 1 \end{array} \right)\), find

1. the angle between the directions of \(\mathbf { a }\) and \(\mathbf { b }\),
2. the value of \(p\) for which \(\mathbf { b }\) and \(\mathbf { c }\) are perpendicular.

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.