1.10c Magnitude and direction: of vectors

[In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal vectors due east and north respectively.] At time \(t = 0\), a football player kicks a ball from the point \(A\) with position vector \((2\mathbf{i} + \mathbf{j})\) m on a horizontal football field. The motion of the ball is modelled as that of a particle moving horizontally with constant velocity \((5\mathbf{i} + 8\mathbf{j}) \text{ m s}^{-1}\). Find

1. the speed of the ball, [2]
2. the position vector of the ball after \(t\) seconds. [2]

The point \(B\) on the field has position vector \((10\mathbf{i} + 7\mathbf{j})\) m.

1. Find the time when the ball is due north of \(B\). [2]

At time \(t = 0\), another player starts running due north from \(B\) and moves with constant speed \(v \text{ m s}^{-1}\). Given that he intercepts the ball,

1. find the value of \(v\). [6]
2. State one physical factor, other than air resistance, which would be needed in a refinement of the model of the ball's motion to make the model more realistic. [1]

A particle is acted upon by two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\), given by \(\mathbf{F}_1 = (\mathbf{i} - 3\mathbf{j})\) N, \(\mathbf{F}_2 = (p\mathbf{i} + 2p\mathbf{j})\) N, where \(p\) is a positive constant.

1. Find the angle between \(\mathbf{F}_2\) and \(\mathbf{j}\). [2]

The resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is \(\mathbf{R}\). Given that \(\mathbf{R}\) is parallel to \(\mathbf{i}\),

1. find the value of \(p\). [4]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively.] A hiker \(H\) is walking with constant velocity \((1.2\mathbf{i} - 0.9\mathbf{j})\) m s\(^{-1}\).

1. Find the speed of \(H\). [2]

\includegraphics{figure_3} A horizontal field \(OABC\) is rectangular with \(OA\) due east and \(OC\) due north, as shown in Figure 3. At twelve noon hiker \(H\) is at the point \(Y\) with position vector \(100\mathbf{j}\) m, relative to the fixed origin \(O\).

1. Write down the position vector of \(H\) at time \(t\) seconds after noon. [2]

At noon, another hiker \(K\) is at the point with position vector \((9\mathbf{i} + 46\mathbf{j})\) m. Hiker \(K\) is moving with constant velocity \((0.75\mathbf{i} + 1.8\mathbf{j})\) m s\(^{-1}\).

1. Show that, at time \(t\) seconds after noon, $$\overrightarrow{HK} = [(9 - 0.45t)\mathbf{i} + (2.7t - 54)\mathbf{j}] \text{ metres.}$$ [4]

Hence,

1. show that the two hikers meet and find the position vector of the point where they meet. [5]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. Position vectors are given relative to a fixed origin \(O\).] Two ships \(P\) and \(Q\) are moving with constant velocities. Ship \(P\) moves with velocity \((2\mathbf{i} - 3\mathbf{j})\) km h\(^{-1}\) and ship \(Q\) moves with velocity \((3\mathbf{i} + 4\mathbf{j})\) km h\(^{-1}\).

1. Find, to the nearest degree, the bearing on which \(Q\) is moving. [2]

At 2 pm, ship \(P\) is at the point with position vector \((\mathbf{i} + \mathbf{j})\) km and ship \(Q\) is at the point with position vector \((-2\mathbf{j})\) km. At time \(t\) hours after 2 pm, the position vector of \(P\) is \(\mathbf{p}\) km and the position vector of \(Q\) is \(\mathbf{q}\) km.

1. Write down expressions, in terms of \(t\), for
   1. \(\mathbf{p}\),
   2. \(\mathbf{q}\),
   3. \(\overrightarrow{PQ}\). [5]
2. Find the time when
   1. \(Q\) is due north of \(P\),
   2. \(Q\) is north-west of \(P\). [4]

[In this question, the horizontal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are directed due east and due north respectively.] The velocity, \(\mathbf{v} \text{ m s}^{-1}\), of a particle \(P\) at time \(t\) seconds is given by $$\mathbf{v} = (1 - 2t)\mathbf{i} + (3t - 3)\mathbf{j}$$

1. Find the speed of \(P\) when \(t = 0\) [3]
2. Find the bearing on which \(P\) is moving when \(t = 2\) [2]
3. Find the value of \(t\) when \(P\) is moving
   1. parallel to \(\mathbf{j}\),
   2. parallel to \((-\mathbf{i} - 3\mathbf{j})\). [6]

A particle \(P\) moves in a horizontal plane. The acceleration of \(P\) is \((-\mathbf{i} + 2\mathbf{j}) \text{ m s}^{-2}\). At time \(t = 0\), the velocity of \(P\) is \((2\mathbf{i} - 3\mathbf{j}) \text{ m s}^{-1}\).

1. Find, to the nearest degree, the angle between the vector \(\mathbf{j}\) and the direction of motion of \(P\) when \(t = 0\). [3]

At time \(t\) seconds, the velocity of \(P\) is \(\mathbf{v} \text{ m s}^{-1}\). Find

1. an expression for \(\mathbf{v}\) in terms of \(t\), in the form \(a\mathbf{i} + b\mathbf{j}\), [2]
2. the speed of \(P\) when \(t = 3\), [3]
3. the time when \(P\) is moving parallel to \(\mathbf{i}\). [2]

Two cars \(A\) and \(B\) are moving on straight horizontal roads with constant velocities. The velocity of \(A\) is \(20 \text{ m s}^{-1}\) due east, and the velocity of \(B\) is \((10\mathbf{i} + 10\mathbf{j}) \text{ m s}^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively. Initially \(A\) is at the fixed origin \(O\), and the position vector of \(B\) is \(300\mathbf{j}\) m relative to \(O\). At time \(t\) seconds, the position vectors of \(A\) and \(B\) are \(\mathbf{r}\) metres and \(\mathbf{s}\) metres respectively.

1. Find expressions for \(\mathbf{r}\) and \(\mathbf{s}\) in terms of \(t\). [3]
2. Hence write down an expression for \(\overrightarrow{AB}\) in terms of \(t\). [1]
3. Find the time when the bearing of \(B\) from \(A\) is \(045°\). [5]
4. Find the time when the cars are again 300 m apart. [6]

A particle \(P\) of mass 2 kg is moving with velocity \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) when it receives an impulse. Immediately after the impulse is applied, \(P\) has velocity \((2\mathbf{i} - 3\mathbf{j})\) m s\(^{-1}\).

1. Find the magnitude of the impulse. [5]
2. Find the angle between the direction of the impulse and the direction of motion of \(P\) immediately before the impulse is applied. [3]

The velocity v m s\(^{-1}\) of a particle \(P\) at time \(t\) seconds is given by $$\mathbf{v} = (3t - 2)\mathbf{i} - 5t\mathbf{j}.$$

1. Show that the acceleration of \(P\) is constant. [2]

At \(t = 0\), the position vector of \(P\) relative to a fixed origin \(O\) is 3i m.

1. Find the distance of \(P\) from \(O\) when \(t = 2\). [6]

A particle \(P\) of mass 0.4 kg is moving so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds is given by $$\mathbf{r} = (t^2 + 4t)\mathbf{i} + (3t - t^3)\mathbf{j}.$$

1. Calculate the speed of \(P\) when \(t = 3\). [5]

When \(t = 3\), the particle \(P\) is given an impulse \((8\mathbf{i} - 12\mathbf{j})\) N s.

1. Find the velocity of \(P\) immediately after the impulse. [3]

\includegraphics{figure_3} [In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) being vertical.] A particle \(P\) is projected from the point \(A\) which has position vector \(47.5\mathbf{j}\) metres with respect to a fixed origin \(O\). The velocity of projection of \(P\) is \((2u\mathbf{i} + 5u\mathbf{j})\) m s\(^{-1}\). The particle moves freely under gravity passing through the point \(B\) with position vector \(30\mathbf{i}\) metres, as shown in Figure 3.

1. Show that the time taken for \(P\) to move from \(A\) to \(B\) is 5 s. [6]
2. Find the value of \(u\). [2]
3. Find the speed of \(P\) at \(B\). [5]

\includegraphics{figure_1} The points \(A\), \(B\) and \(C\) lie in a horizontal plane. A batsman strikes a ball of mass \(0.25\) kg. Immediately before being struck, the ball is moving along the horizontal line \(AB\) with speed \(30 \text{ ms}^{-1}\). Immediately after being struck, the ball moves along the horizontal line \(BC\) with speed \(40 \text{ ms}^{-1}\). The line \(BC\) makes an angle of \(60°\) with the original direction of motion \(AB\), as shown in Figure 1. Find, to 3 significant figures,

1. the magnitude of the impulse given to the ball,
2. the size of the angle that the direction of this impulse makes with the original direction of motion \(AB\).

[8]

A particle \(P\) is moving in a plane. At time \(t\) seconds, \(P\) is moving with velocity \(\mathbf{v}\) m s\(^{-1}\), where \(\mathbf{v} = 2t\mathbf{i} - 3t^2\mathbf{j}\). Find

1. the speed of \(P\) when \(t = 4\) [2]
2. the acceleration of \(P\) when \(t = 4\) [3]

Given that \(P\) is at the point with position vector \((-4\mathbf{i} + \mathbf{j})\) m when \(t = 1\),

1. find the position vector of \(P\) when \(t = 4\) [5]

[In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively.] \includegraphics{figure_3} The point \(O\) is a fixed point on a horizontal plane. A ball is projected from \(O\) with velocity \((6\mathbf{i} + 12\mathbf{j})\) m s\(^{-1}\), and passes through the point \(A\) at time \(t\) seconds after projection. The point \(B\) is on the horizontal plane vertically below \(A\), as shown in Figure 3. It is given that \(OB = 2AB\). Find

1. the value of \(t\), [7]
2. the speed, \(V\) m s\(^{-1}\), of the ball at the instant when it passes through \(A\). [5]

At another point \(C\) on the path the speed of the ball is also \(V\) m s\(^{-1}\).

1. Find the time taken for the ball to travel from \(O\) to \(C\). [3]

\includegraphics{figure_2} The points \(A (3, 0)\) and \(B (0, 4)\) are two vertices of the rectangle \(ABCD\), as shown in Fig. 2.

1. Write down the gradient of \(AB\) and hence the gradient of \(BC\). [3]

The point \(C\) has coordinates \((8, k)\), where \(k\) is a positive constant.

1. Find the length of \(BC\) in terms of \(k\). [2]

Given that the length of \(BC\) is 10 and using your answer to part (b),

1. find the value of \(k\), [4]
2. find the coordinates of \(D\). [2]

\includegraphics{figure_1} Figure 1 shows the cross-section ABCD of a chocolate bar, where AB, CD and AD are straight lines and M is the mid-point of AD. The length AD is 28 mm, and BC is an arc of a circle with centre M. Taking A as the origin, B, C and D have coordinates (7, 24), (21, 24) and (28, 0) respectively.

1. Show that the length of BM is 25 mm. [1]
2. Show that, to 3 significant figures, \(\angle BMC = 0.568\) radians. [3]
3. Hence calculate, in mm², the area of the cross-section of the chocolate bar. [5]

Given that this chocolate bar has length 85 mm,

1. calculate, to the nearest cm³, the volume of the bar. [2]

The line \(l_1\) passes through the point \(A(0, 6, 9)\) and the point \(B(4, -6, -11)\). The line \(l_2\) has equation \(\mathbf{r} = \begin{bmatrix} -1 \\ 5 \\ -2 \end{bmatrix} + \lambda \begin{bmatrix} 3 \\ -5 \\ 1 \end{bmatrix}\).

1. The acute angle between the lines \(l_1\) and \(l_2\) is \(\theta\). Find the value of \(\cos \theta\) as a fraction in its lowest terms. [5 marks]
2. Show that the lines \(l_1\) and \(l_2\) intersect and find the coordinates of the point of intersection. [5 marks]
3. The points \(C\) and \(D\) lie on line \(l_2\) such that \(ACBD\) is a parallelogram. \includegraphics{figure_6} The length of \(AB\) is three times the length of \(CD\). Find the coordinates of the points \(C\) and \(D\). [5 marks]

Referred to an origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \((\mathbf{9i} - \mathbf{2j} + \mathbf{k})\), \((\mathbf{6i} + \mathbf{2j} + \mathbf{6k})\) and \((\mathbf{3i} + p\mathbf{j} + q\mathbf{k})\) respectively, where \(p\) and \(q\) are constants.

1. Find, in vector form, an equation of the line \(l\) which passes through \(A\) and \(B\). [2]

Given that \(C\) lies on \(l\),

1. find the value of \(p\) and the value of \(q\), [2]
2. calculate, in degrees, the acute angle between \(OC\) and \(AB\). [3]

The point \(D\) lies on \(AB\) and is such that \(OD\) is perpendicular to \(AB\).

1. Find the position vector of \(D\). [6]

The points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) relative to an origin \(O\), where \(\mathbf{a} = 4\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}\) and \(\mathbf{b} = -7\mathbf{i} + 5\mathbf{j} + 4\mathbf{k}\).

1. Find the length of \(AB\). [3]
2. Use a scalar product to find angle \(OAB\). [3]

The position vectors of three points \(A\), \(B\) and \(C\) relative to an origin \(O\) are given respectively by $$\overrightarrow{OA} = 7\mathbf{i} + 3\mathbf{j} - 3\mathbf{k},$$ $$\overrightarrow{OB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$$ and $$\overrightarrow{OC} = 5\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the angle between \(AB\) and \(AC\). [6]
2. Find the area of triangle \(ABC\). [2]

A piece of cloth ABDC is attached to the tops of vertical poles AE, BF, DG and CH, where E, F, G and H are at ground level (see Fig. 7). Coordinates are as shown, with lengths in metres. The length of pole DG is \(k\) metres. \includegraphics{figure_7}

1. Write down the vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\). Hence calculate the angle BAC. [6]
2. Verify that the equation of the plane ABC is \(x + y - 2z + d = 0\), where \(d\) is a constant to be determined. Calculate the acute angle the plane makes with the horizontal plane. [7]
3. Given that A, B, D and C are coplanar, show that \(k = 3\). Hence show that ABDC is a trapezium, and find the ratio of CD to AB. [5]

Fig. 7 shows a tetrahedron ABCD. The coordinates of the vertices, with respect to axes Oxyz, are A(-3, 0, 0), B(2, 0, -2), C(0, 4, 0) and D(0, 4, 5). \includegraphics{figure_7}

1. Find the lengths of the edges AB and AC, and the size of the angle CAB. Hence calculate the area of triangle ABC. [7]
2. 1. Verify that 4i - 3j + 10k is normal to the plane ABC. [2]
   2. Hence find the equation of this plane. [2]
3. Write down a vector equation for the line through D perpendicular to the plane ABC. Hence find the point of intersection of this line with the plane ABC. [5]

The volume of a tetrahedron is \(\frac{1}{3} \times \text{area of base} \times \text{height}\).

1. Find the volume of the tetrahedron ABCD. [2]

Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\begin{pmatrix} 1 \\ 5 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 3 \\ -6 \end{pmatrix}\) respectively. Find, in exact, simplified form,

1. the cosine of \(\angle AOB\), [4]
2. the area of triangle \(OAB\), [4]
3. the shortest distance from \(A\) to the line \(OB\). [2]

A straight road passes through villages at the points \(A\) and \(B\) with position vectors \((9\mathbf{i} - 8\mathbf{j} + 2\mathbf{k})\) and \((4\mathbf{j} + \mathbf{k})\) respectively, relative to a fixed origin. The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf{r} = (2\mathbf{i} + 16\mathbf{j} - \mathbf{k}) + t(-5\mathbf{i} + 3\mathbf{j}),$$ where \(t\) is a scalar parameter.

1. Find the position vector of \(C\). [5]

Given that 1 unit on each coordinate axis represents 200 metres,

1. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\). [4]

A bee flies in a straight line from \(A\) to \(B\), where \(\overrightarrow{AB} = (3\mathbf{i} - 12\mathbf{j})\) m, in 5 seconds at a constant speed. Find

1. the straight-line distance \(AB\), [2 marks]
2. the speed of the bee, [2 marks]
3. the velocity vector of the bee. [2 marks]