1.10c Magnitude and direction: of vectors

4. Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a rigid body, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( \mathbf { i } + 4 \mathbf { k } ) \mathrm { N }\). The force \(\mathbf { F } _ { 1 }\) acts through the point with position vector \(( \mathbf { i } + \mathbf { k } ) \mathrm { m }\) relative to a fixed origin \(O\). The force \(\mathbf { F } _ { 2 }\) acts through the point with position vector ( \(2 \mathbf { j }\) ) m . The two forces are equivalent to a single force \(\mathbf { F }\).

1. Find the magnitude of \(\mathbf { F }\).
2. Find, in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), a vector equation of the line of action of \(\mathbf { F }\).

10 A vector \(\mathbf { v }\) has magnitude 1 unit. The angle between \(\mathbf { v }\) and the positive \(z\)-axis is \(60 ^ { \circ }\), and \(\mathbf { v }\) is parallel to the plane \(x - 2 y = 0\). Given that \(\mathbf { v } = a \mathbf { i } + b \mathbf { j } + c \mathbf { k }\), where \(a , b\) and \(c\) are all positive, find \(\mathbf { v }\). \section*{END OF QUESTION PAPER}

2 A particle, Q , moves so that its velocity, \(\mathbf { v }\), at time \(t\) is given by \(\mathbf { v } = ( 6 t - 6 ) \mathbf { i } + \left( 3 - 2 t + t ^ { 2 } \right) \mathbf { j } + 4 \mathbf { k }\), where \(0 \leqslant t \leqslant 6\).

1. Explain how you know that Q is never stationary. When Q is at a point A the direction of the acceleration of Q is parallel to the \(\mathbf { i }\) direction. When Q is at a point B the direction of the acceleration of Q makes an angle of \(45 ^ { \circ }\) with the \(\mathbf { i }\) direction.
2. Determine the straight-line distance AB .

1 A particle, P , has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds given by \(\mathbf { v } = \left( \begin{array} { c } 6 \left( t ^ { 2 } - 3 t + 2 \right) \\ 2 ( 1 - t ) \\ 3 \left( t ^ { 2 } - 1 \right) \end{array} \right)\), where \(0 \leq t \leq 3\).

1. Show that there is just one time at which P is instantaneously at rest and state this value of \(t\). P has a mass of 5 kg and is acted on by a single force \(\mathbf { F }\) N.
2. Find \(\mathbf { F }\) when \(t = 2\).
3. Find an expression for the position, \(\mathbf { r } \mathrm { m }\), of P at time \(t \mathrm {~s}\), given that \(\mathbf { r } = \left( \begin{array} { c } - 5 \\ 2 \\ 6 \end{array} \right)\) when \(t = 0\).

5. A particle of mass 2 kg is moving under the action of a force \(\mathbf { F N }\) which, at time \(t\) seconds, is given by $$\mathbf { F } = 4 t \mathbf { i } - \sqrt { t } \mathbf { j } + 6 \mathbf { k }$$ When \(t = 1\), the velocity of the particle is \(\left( 3 \mathbf { i } - \frac { 1 } { 3 } \mathbf { j } - \mathbf { k } \right) \mathrm { ms } ^ { - 1 }\).

1. Find an expression for the velocity vector of the particle at time \(t \mathrm {~s}\).
2. Determine the values of \(t\) when the particle is moving in a direction perpendicular to the vector \(( - \mathbf { i } + 3 \mathbf { k } )\).

1. Two birds are flying towards their nest, which is in a tree.

Relative to a fixed origin, the flight path of each bird is modelled by a straight line.
In the model, the equation for the flight path of the first bird is $$\mathbf { r } _ { 1 } = \left( \begin{array} { r } - 1 \\ 5 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ a \\ 0 \end{array} \right)$$ and the equation for the flight path of the second bird is $$\mathbf { r } _ { 2 } = \left( \begin{array} { r } 4 \\ - 1 \\ 3 \end{array} \right) + \mu \left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(a\) is a constant.
In the model, the angle between the birds' flight paths is \(120 ^ { \circ }\)

1. Determine the value of \(a\).
2. Verify that, according to the model, there is a common point on the flight paths of the two birds and find the coordinates of this common point. The position of the nest is modelled as being at this common point.
   The tree containing the nest is in a park.
   The ground level of the park is modelled by the plane with equation $$2 x - 3 y + z = 2$$
3. Hence determine the shortest distance from the nest to the ground level of the park.
4. By considering the model, comment on whether your answer to part (c) is reliable, giving a reason for your answer.

1. With respect to a fixed origin \(O\), the line \(l\) has equation

$$( \mathbf { r } - ( 12 \mathbf { i } + 16 \mathbf { j } - 8 \mathbf { k } ) ) \times ( 9 \mathbf { i } + 6 \mathbf { j } + 2 \mathbf { k } ) = \mathbf { 0 }$$ The point \(A\) lies on \(l\) such that the direction cosines of \(\overrightarrow { O A }\) with respect to the \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) axes are \(\frac { 3 } { 7 } , \beta\) and \(\gamma\). Determine the coordinates of the point \(A\).

1. With respect to a fixed origin \(O\), the points \(A\) and \(B\) have coordinates \(( 2,2 , - 1 )\) and ( \(4,2 p , 1\) ) respectively, where \(p\) is a constant.

For each of the following, determine the possible values of \(p\) for which,

1. \(O B\) makes an angle of \(45 ^ { \circ }\) with the positive \(x\)-axis
2. \(\overrightarrow { O A } \times \overrightarrow { O B }\) is parallel to \(\left( \begin{array} { r } 4 \\ - p \\ 2 \end{array} \right)\)
3. the area of triangle \(O A B\) is \(3 \sqrt { 2 }\)

9 In this question you must show detailed reasoning. Find a vector \(\mathbf { v }\) which has the following properties.

• It is a unit vector.
• It is parallel to the plane \(2 x + 2 y + z = 10\).
• It makes an angle of \(45 ^ { \circ }\) with the normal to the plane \(\mathrm { x } + \mathrm { z } = 5\).

\section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
For queries or further information please contact The OCR Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
OCR is part of Cambridge University Press \& Assessment, which is itself a department of the University of Cambridge.

7 The points \(A ( 0,35,120 ) , B ( 28,21,120 )\) and \(C ( 96,35 , - 72 )\) lie on the sphere \(S\), with centre \(O\) and radius 125 . Triangle \(A B C\) is denoted by \(\triangle\).

1. Find, in simplest surd form, the area of \(\Delta\). The points \(A , B\) and \(C\) also form a spherical triangle, \(T\), on the surface of \(S\). Each 'side' of \(T\) is the shorter arc of the circle, centre \(O\) and radius 125, which passes through two of the given vertices of \(T\). In order to find an approximation to the area of the spherical triangle, \(T\) is being modelled by \(\Delta\).
2. (a) State the assumption being made in using this model.
   (b) Say, giving a reason, whether the model gives an under-estimate or an over-estimate of the area of \(T\).

2 Vector \(\mathbf { v } = a \mathbf { i } + 0.6 \mathbf { j }\), where \(a\) is a constant.

1. Given that the direction of \(\mathbf { v }\) is \(45 ^ { \circ }\), state the value of \(a\).
2. Given instead that \(\mathbf { v }\) is parallel to \(8 \mathbf { i } + 3 \mathbf { j }\), find the value of \(a\).
3. Given instead that \(\mathbf { v }\) is a unit vector, find the possible values of \(a\).

3 The points \(P , Q\) and \(R\) have coordinates \(( - 1,6 ) , ( 2,10 )\) and \(( 11,1 )\) respectively. Find the angle \(P R Q\).

6 The points \(A , B\) and \(C\) have coordinates \(( 3 , - 2,4 ) , ( 5,4,0 )\) and \(( 11,6 , - 4 )\) respectively.

1. 1. Find the vector \(\overrightarrow { B A }\).
   2. Show that the size of angle \(A B C\) is \(\cos ^ { - 1 } \left( - \frac { 5 } { 7 } \right)\).
2. The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 8 \\ - 3 \\ 2 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ 3 \\ - 2 \end{array} \right]\).
   1. Verify that \(C\) lies on \(l\).
   2. Show that \(A B\) is parallel to \(l\).
3. The quadrilateral \(A B C D\) is a parallelogram. Find the coordinates of \(D\).

8 The points \(A\) and \(B\) have coordinates \(( 2,1 , - 1 )\) and \(( 3,1 , - 2 )\) respectively. The angle \(O B A\) is \(\theta\), where \(O\) is the origin.

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Show that \(\cos \theta = \frac { 5 } { 2 \sqrt { 7 } }\).
2. The point \(C\) is such that \(\overrightarrow { O C } = 2 \overrightarrow { O B }\). The line \(l\) is parallel to \(\overrightarrow { A B }\) and passes through the point \(C\). Find a vector equation of \(l\).
3. The point \(D\) lies on \(l\) such that angle \(O D C = 90 ^ { \circ }\). Find the coordinates of \(D\).

7 The points \(A\) and \(B\) have coordinates \(( 1,4,2 )\) and \(( 2 , - 1,3 )\) respectively.
The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 2 \\ - 1 \\ 3 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right]\).

1. Show that the distance between the points \(A\) and \(B\) is \(3 \sqrt { 3 }\).
2. The line \(A B\) makes an acute angle \(\theta\) with \(l\). Show that \(\cos \theta = \frac { 7 } { 9 }\).
3. The point \(P\) on the line \(l\) is where \(\lambda = p\).
   1. Show that $$\overrightarrow { A P } \cdot \left[ \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right] = 7 + 3 p$$
   2. Hence find the coordinates of the foot of the perpendicular from the point \(A\) to the line \(l\).

3 A triangle ABC has vertices \(\mathrm { A } ( - 2,4,1 ) , \mathrm { B } ( 2,3,4 )\) and \(\mathrm { C } ( 4,8,3 )\). By calculating a suitable scalar product, show that angle ABC is a right angle. Hence calculate the area of the triangle.

1 The point A has coordinates \(( 2,5,4 )\) and the line BC has equation $$\mathbf { r } = \left( \begin{array} { c } 8 \\ 25 \\ 43 \end{array} \right) + \lambda \left( \begin{array} { c } 4 \\ 15 \\ 25 \end{array} \right)$$ You are given that \(\mathrm { AB } = \mathrm { AC } = 15\).

1. Show that the coordinates of one of the points B and C are (4, 10, 18). Find the coordinates of the other point. These points are B and C respectively.
2. Find the equation of the plane ABC in cartesian form.
3. Show that the plane containing the line BC and perpendicular to the plane ABC has equation \(5 y - 3 z + 4 = 0\). The point D has coordinates \(( 1,1,3 )\).
4. Show that \(| \overrightarrow { B C } \times \overrightarrow { A D } | = \sqrt { 7667 }\) and hence find the shortest distance between the lines \(B C\) and \(A D\).
5. Find the volume of the tetrahedron ABCD .

2 The points \(A\), \(B\) and \(C\) have position vectors \(3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } , - \mathbf { i } + 6 \mathbf { k }\) and \(7 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k }\) respectively. M is the midpoint of BC .

1. Show that the magnitude of \(\overrightarrow { O M }\) is equal to \(\sqrt { 17 }\). Point D is such that \(\overrightarrow { B C } = \overrightarrow { A D }\).
2. Show that position vector of the point D is \(11 \mathbf { i } - 8 \mathbf { j } - 6 \mathbf { k }\).

2 The points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j }\) and \(4 \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k }\) respectively.

1. Find the length of \(A B\). Point \(P\) has position vector \(p \mathbf { i } - 3 \mathbf { k }\), where \(p\) is a constant. \(P\) lies on the circumference of a circle of which \(A B\) is a diameter.
2. Find the two possible values of \(p\).

7 Three points \(A , B\) and \(C\) have coordinates \(A ( 8,17 ) , B ( 15,10 )\) and \(C ( - 2 , - 7 )\) 7

1. Show that angle \(A B C\) is a right angle.
   7
2. \(\quad A , B\) and \(C\) lie on a circle.
   7 (b) (i) Explain why \(A C\) is a diameter of the circle.
   7 (b) (ii) Determine whether the point \(D ( - 8 , - 2 )\) lies inside the circle, on the circle or outside the circle. Fully justify your answer.

12 A particle has a speed of \(6 \mathrm {~ms} ^ { - 1 }\) in a direction relative to unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-18_307_542_1528_749} The velocity of this particle can be expressed as a vector \(\left[ \begin{array} { l } v _ { 1 } \\ v _ { 2 } \end{array} \right] \mathrm { ms } ^ { - 1 }\) Find the correct expression for \(v _ { 2 }\) Circle your answer.
[0pt] [1 mark] \(v _ { 2 } = 6 \cos 30 ^ { \circ }\) \(v _ { 2 } = 6 \sin 30 ^ { \circ }\) \(v _ { 2 } = - 6 \sin 30 ^ { \circ }\) \(v _ { 2 } = - 6 \cos 30 ^ { \circ }\)