1.10c Magnitude and direction: of vectors

9 \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-14_670_857_260_644} The diagram shows a pyramid \(O A B C D\) with a horizontal rectangular base \(O A B C\). The sides \(O A\) and \(A B\) have lengths of 8 units and 6 units respectively. The point \(E\) on \(O B\) is such that \(O E = 2\) units. The point \(D\) of the pyramid is 7 units vertically above \(E\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O C\) and \(E D\) respectively.

1. Show that \(\overrightarrow { O E } = 1.6 \mathbf { i } + 1.2 \mathbf { j }\).
2. Use a scalar product to find angle \(B D O\).

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 \(B P C\).

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 \(O A B\).

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.

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 \(\overrightarrow { O C }\).

4 Relative to an origin \(O\), the position vectors of points \(P\) and \(Q\) are given by $$\overrightarrow { O P } = \left( \begin{array} { r } - 2 \\ 3 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O Q } = \left( \begin{array} { l } 2 \\ 1 \\ q \end{array} \right)$$ where \(q\) is a constant.

1. In the case where \(q = 3\), use a scalar product to show that \(\cos P O Q = \frac { 1 } { 7 }\).
2. Find the values of \(q\) for which the length of \(\overrightarrow { P Q }\) is 6 units.
   \includegraphics[max width=\textwidth, alt={}]{933cdfe1-27bb-450d-8b9a-b494916242cb-3_647_741_845_699}
   The diagram shows the cross-section of a hollow cone and a circular cylinder. The cone has radius 6 cm and height 12 cm , and the cylinder has radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The cylinder just fits inside the cone with all of its upper edge touching the surface of the cone.

4 The position vectors of points \(A\) and \(B\) are \(\left( \begin{array} { r } - 3 \\ 6 \\ 3 \end{array} \right)\) and \(\left( \begin{array} { r } - 1 \\ 2 \\ 4 \end{array} \right)\) respectively, relative to an origin \(O\).

1. Calculate angle \(A O B\).
2. The point \(C\) is such that \(\overrightarrow { A C } = 3 \overrightarrow { A B }\). Find the unit vector in the direction of \(\overrightarrow { O C }\).

10 \includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-4_597_693_274_726} The diagram shows a cube \(O A B C D E F G\) in which the length of each side is 4 units. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively. The mid-points of \(O A\) and \(D G\) are \(P\) and \(Q\) respectively and \(R\) is the centre of the square face \(A B F E\).

1. Express each of the vectors \(\overrightarrow { P R }\) and \(\overrightarrow { P Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to find angle \(Q P R\).
3. Find the perimeter of triangle \(P Q R\), giving your answer correct to 1 decimal place.

4 \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-2_558_1488_863_331} The diagram shows a semicircular prism with a horizontal rectangular base \(A B C D\). The vertical ends \(A E D\) and \(B F C\) are semicircles of radius 6 cm . The length of the prism is 20 cm . The mid-point of \(A D\) is the origin \(O\), the mid-point of \(B C\) is \(M\) and the mid-point of \(D C\) is \(N\). The points \(E\) and \(F\) are the highest points of the semicircular ends of the prism. The point \(P\) lies on \(E F\) such that \(E P = 8 \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O D , O M\) and \(O E\) respectively.

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

8 The equation of a curve is \(y = 5 - \frac { 8 } { x }\).

1. Show that the equation of the normal to the curve at the point \(P ( 2,1 )\) is \(2 y + x = 4\). This normal meets the curve again at the point \(Q\).
2. Find the coordinates of \(Q\).
3. Find the length of \(P Q\).

8 The first term of an arithmetic progression is 8 and the common difference is \(d\), where \(d \neq 0\). The first term, the fifth term and the eighth term of this arithmetic progression are the first term, the second term and the third term, respectively, of a geometric progression whose common ratio is \(r\).

1. Write down two equations connecting \(d\) and \(r\). Hence show that \(r = \frac { 3 } { 4 }\) and find the value of \(d\).
2. Find the sum to infinity of the geometric progression.
3. Find the sum of the first 8 terms of the arithmetic progression.

5

1. Prove the identity \(( \sin x + \cos x ) ( 1 - \sin x \cos x ) \equiv \sin ^ { 3 } x + \cos ^ { 3 } x\).
2. Solve the equation \(( \sin x + \cos x ) ( 1 - \sin x \cos x ) = 9 \sin ^ { 3 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

4

1. Prove the identity \(\frac { \sin x \tan x } { 1 - \cos x } \equiv 1 + \frac { 1 } { \cos x }\).
2. Hence solve the equation \(\frac { \sin x \tan x } { 1 - \cos x } + 2 = 0\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

8 \includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-3_613_897_1311_623} The diagram shows part of the curve \(y = \frac { 2 } { 1 - x }\) and the line \(y = 3 x + 4\). The curve and the line meet at points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\).
2. Find the length of the line \(A B\) and the coordinates of the mid-point of \(A B\).

9 \includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-4_582_1072_255_541} The diagram shows a pyramid \(O A B C P\) in which the horizontal base \(O A B C\) is a square of side 10 cm and the vertex \(P\) is 10 cm vertically above \(O\). The points \(D , E , F , G\) lie on \(O P , A P , B P , C P\) respectively and \(D E F G\) is a horizontal square of side 6 cm . The height of \(D E F G\) above the base is \(a \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively.

1. Show that \(a = 4\).
2. Express the vector \(\overrightarrow { B G }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Use a scalar product to find angle \(G B A\).

10 \includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-4_561_599_744_774} The diagram shows triangle \(O A B\), in which the position vectors of \(A\) and \(B\) with respect to \(O\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = - 3 \mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k } .$$ \(C\) is a point on \(O A\) such that \(\overrightarrow { O C } = p \overrightarrow { O A }\), where \(p\) is a constant.

1. Find angle \(A O B\).
2. Find \(\overrightarrow { B C }\) in terms of \(p\) and vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Find the value of \(p\) given that \(B C\) is perpendicular to \(O A\).

8 Relative to an origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } + 7 \mathbf { j } - p \mathbf { k }\) and the point \(B\) has position vector \(8 \mathbf { i } - \mathbf { j } - p \mathbf { k }\), where \(p\) is a constant.

1. Find \(\overrightarrow { O A } \cdot \overrightarrow { O B }\).
2. Hence show that there are no real values of \(p\) for which \(O A\) and \(O B\) are perpendicular to each other.
3. Find the values of \(p\) for which angle \(A O B = 60 ^ { \circ }\).

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

1. Find the value of \(p\) for which angle \(A O B\) is \(90 ^ { \circ }\).
2. In the case where \(p = 4\), find the vector which has magnitude 28 and is in the same direction as \(\overrightarrow { A B }\).

9 The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are \(\mathbf { a }\) and \(\mathbf { b }\) respectively. The position vectors of points \(C\) and \(D\) relative to \(O\) are \(3 \mathbf { a }\) and \(2 \mathbf { b }\) respectively. It is given that $$\mathbf { a } = \left( \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right) \quad \text { and } \quad \mathbf { b } = \left( \begin{array} { l } 4 \\ 0 \\ 6 \end{array} \right) .$$

1. Find the unit vector in the direction of \(\overrightarrow { C D }\).
2. The point \(E\) is the mid-point of \(C D\). Find angle \(E O D\).

7 The position vectors of the points \(A\) and \(B\), relative to an origin \(O\), are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 0 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } k \\ - k \\ 2 k \end{array} \right)$$ where \(k\) is a constant.

1. In the case where \(k = 2\), calculate angle \(A O B\).
2. Find the values of \(k\) for which \(\overrightarrow { A B }\) is a unit vector.

9 The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { c } p \\ 1 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { l } 4 \\ 2 \\ p \end{array} \right)$$ where \(p\) is a constant.

1. In the case where \(O A B\) is a straight line, state the value of \(p\) and find the unit vector in the direction of \(\overrightarrow { O A }\).
2. In the case where \(O A\) is perpendicular to \(A B\), find the possible values of \(p\).
3. In the case where \(p = 3\), the point \(C\) is such that \(O A B C\) is a parallelogram. Find the position vector of \(C\).

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

1. In the case where \(p = 6\), find the unit vector in the direction of \(\overrightarrow { A B }\).
2. Find the values of \(p\) for which angle \(A O B = \cos ^ { - 1 } \left( \frac { 1 } { 5 } \right)\).