1.10d Vector operations: addition and scalar multiplication

\includegraphics{figure_7} The diagram shows a solid cylinder standing on a horizontal circular base with centre \(O\) and radius \(4\) units. Points \(A\), \(B\) and \(C\) lie on the circumference of the base such that \(AB\) is a diameter and angle \(BOC = 90°\). Points \(P\), \(Q\) and \(R\) lie on the upper surface of the cylinder vertically above \(A\), \(B\) and \(C\) respectively. The height of the cylinder is \(12\) units. The mid-point of \(CR\) is \(M\) and \(N\) lies on \(BQ\) with \(BN = 4\) units. Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OB\) and \(OC\) respectively and the unit vector \(\mathbf{k}\) is vertically upwards. Evaluate \(\overrightarrow{PN} \cdot \overrightarrow{PM}\) and hence find angle \(MPN\). [7]

With respect to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors given by $$\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 0 \\ 4 \\ 1 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} -3 \\ -2 \\ 2 \end{pmatrix}.$$

1. The point \(D\) is such that \(ABCD\) is a trapezium with \(\overrightarrow{DC} = 3\overrightarrow{AB}\). Find the position vector of \(D\). [2]
2. The diagonals of the trapezium intersect at the point \(P\). Find the position vector of \(P\). [5]
3. Using a scalar product, calculate angle \(ABC\). [4]

\includegraphics{figure_5} The diagram shows a three-dimensional shape. The base \(OAB\) is a horizontal triangle in which angle \(AOB\) is 90°. The side \(OBCD\) is a rectangle and the side \(OAD\) lies in a vertical plane. Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OA\) and \(OB\) respectively and the unit vector \(\mathbf{k}\) is vertical. The position vectors of \(A\), \(B\) and \(D\) are given by \(\overrightarrow{OA} = 8\mathbf{i}\), \(\overrightarrow{OB} = 5\mathbf{j}\) and \(\overrightarrow{OD} = 2\mathbf{i} + 4\mathbf{k}\).

1. Express each of the vectors \(\overrightarrow{DA}\) and \(\overrightarrow{CA}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [2]
2. Use a scalar product to find angle \(CAD\). [4]

\includegraphics{figure_9} The diagram shows a pyramid \(OABCD\) with a horizontal rectangular base \(OABC\). The sides \(OA\) and \(AB\) have lengths of 8 units and 6 units respectively. The point \(E\) on \(OB\) is such that \(OE = 2\) 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 \(OA\), \(OC\) and \(ED\) respectively.

1. Show that \(\overrightarrow{OE} = 1.6\mathbf{i} + 1.2\mathbf{j}\). [2]
2. Use a scalar product to find angle \(BDO\). [7]

\includegraphics{figure_9} The diagram shows three points \(A\), \(B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow{OA} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}\), \(\overrightarrow{OB} = \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}\) and \(\overrightarrow{OC} = \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}\). The point \(D\) lies on \(BC\), between \(B\) and \(C\), and is such that \(CD = 2DB\).

1. Find the equation of the plane \(ABC\), giving your answer in the form \(ax + by + cz = d\). [6]
2. Find the position vector of \(D\). [1]
3. Show that the length of the perpendicular from \(A\) to \(OD\) is \(\frac{1}{3}\sqrt{(65)}\). [4]

With respect to a fixed origin, \(O\), the point \(A\) has position vector $$\overrightarrow{OA} = \begin{pmatrix} 7 \\ 2 \\ -5 \end{pmatrix}$$ Given that $$\overrightarrow{AB} = \begin{pmatrix} -2 \\ 4 \\ 3 \end{pmatrix}$$

1. find the coordinates of the point \(B\). [2]

The point \(C\) has position vector $$\overrightarrow{OC} = \begin{pmatrix} a \\ 5 \\ -1 \end{pmatrix}$$ where \(a\) is a constant. Given that \(\overrightarrow{OC}\) is perpendicular to \(\overrightarrow{BC}\)

1. find the possible values of \(a\). [4]

With respect to a fixed origin \(O\), the line \(l_1\) is given by the equation $$\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + \lambda(8\mathbf{i} - \mathbf{j} + 4\mathbf{k})$$ where \(\lambda\) is a scalar parameter. The point \(A\) lies on \(l_1\) Given that \(|\overrightarrow{OA}| = 5\sqrt{10}\)

1. show that at \(A\) the parameter \(\lambda\) satisfies $$81\lambda^2 + 52\lambda - 220 = 0$$ [3]

Hence

1. 1. show that one possible position vector for \(A\) is \(-15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\)
   2. find the other possible position vector for \(A\). [3]

The line \(l_2\) is parallel to \(l_1\) and passes through \(O\). Given that • \(\overrightarrow{OA} = -15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\) • point \(B\) lies on \(l_2\) where \(|\overrightarrow{OB}| = 4\sqrt{10}\)

1. find the area of triangle \(OAB\), giving your answer to one decimal place. [4]

\includegraphics{figure_1} Figure 1 shows a sketch of triangle \(PQR\). Given that • \(\overrightarrow{PQ} = 2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}\) • \(\overrightarrow{PR} = 8\mathbf{i} - 5\mathbf{j} + 3\mathbf{k}\)

1. Find \(\overrightarrow{RQ}\) [2]
2. Find the size of angle \(PQR\), in degrees, to three significant figures. [3]

The line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}$$ and the line \(l_2\) has vector equation $$\mathbf{r} = \begin{pmatrix} 0 \\ 4 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix},$$ where \(\lambda\) and \(\mu\) are parameters. The lines \(l_1\) and \(l_2\) intersect at the point \(B\) and the acute angle between \(l_1\) and \(l_2\) is \(\theta\).

1. Find the coordinates of \(B\). [4]
2. Find the value of \(\cos \theta\), giving your answer as a simplified fraction. [4]

The point \(A\), which lies on \(l_1\), has position vector \(\mathbf{a} = 3\mathbf{i} + \mathbf{j} + 2\mathbf{k}\). The point \(C\), which lies on \(l_2\), has position vector \(\mathbf{c} = 5\mathbf{i} - \mathbf{j} - 2\mathbf{k}\). The point \(D\) is such that \(ABCD\) is a parallelogram.

1. Show that \(|\overrightarrow{AB}| = |\overrightarrow{BC}|\). [3]
2. Find the position vector of the point \(D\). [2]

With respect to a fixed origin \(O\), the line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} -9 \\ 8 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ -4 \\ -3 \end{pmatrix}$$ where \(\mu\) is a scalar parameter. The point \(A\) is on \(l_1\) where \(\mu = 2\).

1. Write down the coordinates of \(A\). [1] The acute angle between \(OA\) and \(l_1\) is \(\theta\), where \(O\) is the origin.
2. Find the value of \(\cos \theta\). [3] The point \(B\) is such that \(\overrightarrow{OB} = 3\overrightarrow{OA}\). The line \(l_2\) passes through the point \(B\) and is parallel to the line \(l_1\).
3. Find a vector equation of \(l_2\). [2]
4. Find the length of \(OB\), giving your answer as a simplified surd. [1] The point \(X\) lies on \(l_2\). Given that the vector \(\overrightarrow{OX}\) is perpendicular to \(l_2\),
5. find the length of \(OX\), giving your answer to 3 significant figures. [3]

With respect to a fixed origin \(O\), the lines \(l_1\) and \(l_2\) are given by the equations $$l_1: \mathbf{r} = \begin{pmatrix} 5 \\ -3 \\ p \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 1 \\ -3 \end{pmatrix}, \quad l_2: \mathbf{r} = \begin{pmatrix} 8 \\ 5 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant. The lines \(l_1\) and \(l_2\) intersect at the point \(A\).

1. Find the coordinates of \(A\). [2]
2. Find the value of the constant \(p\). [3]
3. Find the acute angle between \(l_1\) and \(l_2\), giving your answer in degrees to 2 decimal places. [3]

The point \(B\) lies on \(l_2\) where \(\mu = 1\)

1. Find the shortest distance from the point \(B\) to the line \(l_1\), giving your answer to 3 significant figures. [3]

Relative to a fixed origin \(O\), the point \(A\) has position vector \(4\mathbf{i} + 8\mathbf{j} - \mathbf{k}\), and the point \(B\) has position vector \(7\mathbf{i} + 14\mathbf{j} + 5\mathbf{k}\).

1. Find the vector \(\overrightarrow{AB}\). [1]
2. Calculate the cosine of \(\angle OAB\). [3]
3. Show that, for all values of \(\lambda\), the point \(P\) with position vector $$\lambda\mathbf{i} + 2\lambda\mathbf{j} + (2\lambda - 9)\mathbf{k}$$ lies on the line through \(A\) and \(B\). [3]
4. Find the value of \(\lambda\) for which \(OP\) is perpendicular to \(AB\). [3]
5. Hence find the coordinates of the foot of the perpendicular from \(O\) to \(AB\). [2]

The points \(A\), \(B\), \(C\), and \(D\) have position vectors $$\mathbf{a} = 2\mathbf{i} + \mathbf{k}, \quad \mathbf{b} = \mathbf{i} + 3\mathbf{j}, \quad \mathbf{c} = \mathbf{i} + 3\mathbf{j} + 2\mathbf{k}, \quad \mathbf{d} = 4\mathbf{j} + \mathbf{k}$$ respectively.

1. Find \(\overrightarrow{AB} \times \overrightarrow{AC}\) and hence find the area of triangle \(ABC\). [7]
2. Find the volume of the tetrahedron \(ABCD\). [2]
3. Find the perpendicular distance of \(D\) from the plane containing \(A\), \(B\) and \(C\). [3]

(Total 12 marks)

A particle \(P\), of mass 3 kg, moves under the action of two constant forces (6\(\mathbf{i}\) + 2\(\mathbf{j}\)) N and (3\(\mathbf{i}\) - 5\(\mathbf{j}\)) N.

1. Find, in the form (\(a\mathbf{i}\) + \(b\mathbf{j}\)) N, the resultant force \(\mathbf{F}\) acting on \(P\). [1]
2. Find, in degrees to one decimal place, the angle between \(\mathbf{F}\) and \(\mathbf{j}\). [3]
3. Find the acceleration of \(P\), giving your answer as a vector. [2]

The initial velocity of \(P\) is (-2\(\mathbf{i}\) + \(\mathbf{j}\)) m s\(^{-1}\).

1. Find, to 3 significant figures, the speed of \(P\) after 2 s. [5]

A particle \(P\) of mass 0.4 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. Initially the velocity of \(P\) is \((6\mathbf{i} - 2\mathbf{j})\) m s\(^{-1}\) and 4 s later the velocity of \(P\) is \((-14\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\).

1. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the acceleration of \(P\). [3]
2. Calculate the magnitude of \(\mathbf{F}\). [3]

Two ships \(P\) and \(Q\) are moving along straight lines with constant velocities. Initially \(P\) is at a point \(O\) and the position vector of \(Q\) relative to \(O\) is \((6\mathbf{i} + 12\mathbf{j})\) km, where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively. The ship \(P\) is moving with velocity \(10\mathbf{j}\) km h\(^{-1}\) and \(Q\) is moving with velocity \((-8\mathbf{i} + 6\mathbf{j})\) km h\(^{-1}\). At time \(t\) hours the position vectors of \(P\) and \(Q\) relative to \(O\) are \(\mathbf{p}\) km and \(\mathbf{q}\) km respectively.

1. Find \(\mathbf{p}\) and \(\mathbf{q}\) in terms of \(t\). [3]
2. Calculate the distance of \(Q\) from \(P\) when \(t = 3\). [3]
3. Calculate the value of \(t\) when \(Q\) is due north of \(P\). [2]

[In this question the vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors in the direction due east and due north respectively.] Two boats \(A\) and \(B\) are moving with constant velocities. Boat \(A\) moves with velocity \(9\mathbf{j}\) km h\(^{-1}\). Boat \(B\) moves with velocity \((3\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\).

1. Find the bearing on which \(B\) is moving. [2]

At noon, \(A\) is at point \(O\), and \(B\) is 10 km due west of \(O\). At time \(t\) hours after noon, the position vectors of \(A\) and \(B\) relative to \(O\) are \(\mathbf{a}\) km and \(\mathbf{b}\) km respectively.

1. Find expressions for \(\mathbf{a}\) and \(\mathbf{b}\) in terms of \(t\), giving your answer in the form \(p\mathbf{i} + q\mathbf{j}\). [3]
2. Find the time when \(B\) is due south of \(A\). [2]

At time \(t\) hours after noon, the distance between \(A\) and \(B\) is \(d\) km. By finding an expression for \(\overrightarrow{AB}\),

1. show that \(d^2 = 25t^2 - 60t + 100\). [4]

At noon, the boats are 10 km apart.

1. Find the time after noon at which the boats are again 10 km apart. [3]

Two ships \(P\) and \(Q\) are travelling at night with constant velocities. At midnight, \(P\) is at the point with position vector \((20\mathbf{i} + 10\mathbf{j})\) km relative to a fixed origin \(O\). At the same time, \(Q\) is at the point with position vector \((14\mathbf{i} - 6\mathbf{j})\) km. Three hours later, \(P\) is at the point with position vector \((29\mathbf{i} + 34\mathbf{j})\) km. The ship \(Q\) travels with velocity \(12\mathbf{j}\) km h\(^{-1}\). At time \(t\) hours after midnight, the position vectors of \(P\) and \(Q\) are \(\mathbf{p}\) km and \(\mathbf{q}\) km respectively. Find

1. the velocity of \(P\), in terms of \(\mathbf{i}\) and \(\mathbf{j}\), [2]
2. expressions for \(\mathbf{p}\) and \(\mathbf{q}\), in terms of \(t\), \(\mathbf{i}\) and \(\mathbf{j}\). [4]

At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d\) km.

1. By finding an expression for \(\overrightarrow{PQ}\), show that $$d^2 = 25t^2 - 92t + 292.$$ [5]

Weather conditions are such that an observer on \(P\) can only see the lights on \(Q\) when the distance between \(P\) and \(Q\) is 15 km or less. Given that when \(t = 1\), the lights on \(Q\) move into sight of the observer,

1. find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer. [5]

Two forces \(\mathbf{P}\) and \(\mathbf{Q}\) act on a particle. The force \(\mathbf{P}\) has magnitude \(7\) N and acts due north. The resultant of \(\mathbf{P}\) and \(\mathbf{Q}\) is a force of magnitude \(10\) N acting in a direction with bearing \(120°\). Find

1. the magnitude of \(\mathbf{Q}\),
2. the direction of \(\mathbf{Q}\), giving your answer as a bearing.

[9]

[In this question the horizontal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are due east and due north respectively.] A model boat \(A\) moves on a lake with constant velocity \((-\mathbf{i} + 6\mathbf{j}) \text{ m s}^{-1}\). At time \(t = 0\), \(A\) is at the point with position vector \((2\mathbf{i} - 10\mathbf{j})\) m. Find

1. the speed of \(A\), [2]
2. the direction in which \(A\) is moving, giving your answer as a bearing. [3]

At time \(t = 0\), a second boat \(B\) is at the point with position vector \((-26\mathbf{i} + 4\mathbf{j})\) m. Given that the velocity of \(B\) is \((3\mathbf{i} + 4\mathbf{j}) \text{ m s}^{-1}\),

1. show that \(A\) and \(B\) will collide at a point \(P\) and find the position vector of \(P\). [5]

Given instead that \(B\) has speed \(8 \text{ m s}^{-1}\) and moves in the direction of the vector \((3\mathbf{i} + 4\mathbf{j})\),

1. find the distance of \(B\) from \(P\) when \(t = 7\) s. [6]

[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship \(S\) is moving along a straight line with constant velocity. At time \(t\) hours the position vector of \(S\) is \(\mathbf{s}\) km. When \(t = 0\), \(\mathbf{s} = 9\mathbf{i} - 6\mathbf{j}\). When \(t = 4\), \(\mathbf{s} = 21\mathbf{i} + 10\mathbf{j}\). Find

1. the speed of \(S\), [4]
2. the direction in which \(S\) is moving, giving your answer as a bearing. [2]
3. Show that \(\mathbf{s} = (3t + 9)\mathbf{i} + (4t - 6)\mathbf{j}\). [2]

A lighthouse \(L\) is located at the point with position vector \((18\mathbf{i} + 6\mathbf{j})\) km. When \(t = T\), the ship \(S\) is 10 km from \(L\).

1. Find the possible values of \(T\). [6]

[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship sets sail at 9 am from a port \(P\) and moves with constant velocity. The position vector of \(P\) is \((4\mathbf{i} - 8\mathbf{j})\) km. At 9.30 am the ship is at the point with position vector \((\mathbf{i} - 4\mathbf{j})\) km.

1. Find the speed of the ship in km h\(^{-1}\). [4]
2. Show that the position vector \(\mathbf{r}\) km of the ship, \(t\) hours after 9 am, is given by \(\mathbf{r} = (4 - 6t)\mathbf{i} + (8t - 8)\mathbf{j}\). [2]

At 10 am, a passenger on the ship observes that a lighthouse \(L\) is due west of the ship. At 10.30 am, the passenger observes that \(L\) is now south-west of the ship.

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

A particle \(P\) of mass \(2 \text{ kg}\) moves in a plane under the action of a single constant force \(\mathbf{F}\) newtons. At time \(t\) seconds, the velocity of \(P\) is \(\mathbf{v} \text{ m s}^{-1}\). When \(t = 0\), \(\mathbf{v} = (-5\mathbf{i} + 7\mathbf{j})\) and when \(t = 3\), \(\mathbf{v} = (\mathbf{i} - 2\mathbf{j})\).

1. Find in degrees the angle between the direction of motion of \(P\) when \(t = 3\) and the vector \(\mathbf{j}\). [3]
2. Find the acceleration of \(P\). [2]
3. Find the magnitude of \(\mathbf{F}\). [3]
4. Find in terms of \(t\) the velocity of \(P\). [2]
5. Find the time at which \(P\) is moving parallel to the vector \(\mathbf{i} + \mathbf{j}\). [3]

A small boat \(S\), drifting in the sea, is modelled as a particle moving in a straight line at constant speed. When first sighted at 0900, \(S\) is at a point with position vector \((4\mathbf{i} - 6\mathbf{j})\) km relative to a fixed origin \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. At 0945, \(S\) is at the point with position vector \((7\mathbf{i} - 7.5\mathbf{j})\) km. At time \(t\) hours after 0900, \(S\) is at the point with position vector \(\mathbf{s}\) km.

1. Calculate the bearing on which \(S\) is drifting. [4]
2. Find an expression for \(\mathbf{s}\) in terms of \(t\). [3]

At 1000 a motor boat \(M\) leaves \(O\) and travels with constant velocity \((p\mathbf{i} + q\mathbf{j})\) km h\(^{-1}\). Given that \(M\) intercepts \(S\) at 1015,

1. calculate the value of \(p\) and the value of \(q\). [6]

[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]