1.10b Vectors in 3D: i,j,k notation

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

1. Find the length of \(A B\).
2. Use a scalar product to find angle \(O A B\).

7 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) respectively, where \(\mathbf { a } = 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k }\) and \(\mathbf { b } = - \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\). The lines \(L _ { 1 }\) and \(L _ { 2 }\) have the vector equations $$\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } , \quad \mathbf { r } = 2 \mathbf { b } + \mu \mathbf { a }$$ respectively.

1. Determine whether or not \(L _ { 1 }\) and \(L _ { 2 }\) intersect.
2. Find the acute angle between the directions of \(L _ { 1 }\) and \(L _ { 2 }\). The point \(C\) has position vector \(\mathbf { c } = p \mathbf { i } + \mathbf { j } + r \mathbf { k }\).
3. Given that \(O C\) is perpendicular to the triangle \(O A B\), determine \(p\) and \(r\).
4. Determine the volume of the tetrahedron \(O A B C\).

Relative to a fixed origin \(O\), the point \(A\) has position vector \(\begin{pmatrix} -2 \\ 4 \\ 7 \end{pmatrix}\) and the point \(B\) has position vector \(\begin{pmatrix} -1 \\ 3 \\ 8 \end{pmatrix}\) The line \(l_1\) passes through the points \(A\) and \(B\).

1. [(a)] Find the vector \(\overrightarrow{AB}\). \hfill [2]
2. [(b)] Hence find a vector equation for the line \(l_1\) \hfill [1]

The point \(P\) has position vector \(\begin{pmatrix} 0 \\ 2 \\ 3 \end{pmatrix}\) Given that angle \(PBA\) is \(\theta\),

1. [(c)] show that \(\cos\theta = \frac{1}{3}\) \hfill [3]

The line \(l_2\) passes through the point \(P\) and is parallel to the line \(l_1\)

1. [(d)] Find a vector equation for the line \(l_2\) \hfill [2]

The points \(C\) and \(D\) both lie on the line \(l_2\) Given that \(AB = PC = DP\) and the \(x\) coordinate of \(C\) is positive,

1. [(e)] find the coordinates of \(C\) and the coordinates of \(D\). \hfill [3]
2. [(f)] find the exact area of the trapezium \(ABCD\), giving your answer as a simplified surd. \hfill [4] \end{enumerate} \end{enumerate} \end{enumerate} \end{enumerate}

\includegraphics{figure_6} The diagram shows a solid figure \(ABCDEF\) in which the horizontal base \(ABC\) is a triangle right-angled at \(A\). The lengths of \(AB\) and \(AC\) are 8 units and 4 units respectively and \(M\) is the mid-point of \(AB\). The point \(D\) is 7 units vertically above \(A\). Triangle \(DEF\) lies in a horizontal plane with \(DE\), \(DF\) and \(FE\) parallel to \(AB\), \(AC\) and \(CB\) respectively and \(N\) is the mid-point of \(FE\). The lengths of \(DE\) and \(DF\) are 4 units and 2 units respectively. Unit vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are parallel to \(\overrightarrow{AB}\), \(\overrightarrow{AC}\) and \(\overrightarrow{AD}\) respectively.

1. Find \(\overrightarrow{MF}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [1]
2. Find \(\overrightarrow{FN}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [1]
3. Find \(\overrightarrow{MN}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [1]
4. Use a scalar product to find angle \(FMN\). [4]

\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]

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]

The line \(l_1\) has vector equation \(\mathbf{r} = \begin{pmatrix} 11 \\ 5 \\ 6 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ 2 \\ 4 \end{pmatrix}\), where \(\lambda\) is a parameter. The line \(l_2\) has vector equation \(\mathbf{r} = \begin{pmatrix} 24 \\ 4 \\ 13 \end{pmatrix} + \mu \begin{pmatrix} 7 \\ 1 \\ 5 \end{pmatrix}\), where \(\mu\) is a parameter.

1. Show that the lines \(l_1\) and \(l_2\) intersect. [4]
2. Find the coordinates of their point of intersection. [2]

Given that \(\theta\) is the acute angle between \(l_1\) and \(l_2\),

1. find the value of \(\cos \theta\). Give your answer in the form \(k\sqrt{3}\), where \(k\) is a simplified fraction. [4]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively.] A particle \(P\) is moving with constant velocity \((-6\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\). At time \(t = 0\), \(P\) passes through the point with position vector \((21\mathbf{i} + 5\mathbf{j})\) m, relative to a fixed origin \(O\).

1. Find the direction of motion of \(P\), giving your answer as a bearing to the nearest degree. [3]
2. Write down the position vector of \(P\) at time \(t\) seconds. [1]
3. Find the time at which \(P\) is north-west of \(O\). [3]

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]

[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]

A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. When \(t = 0\), \(P\) has velocity \((3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and at time \(t = 4\) s, \(P\) has velocity \((15\mathbf{i} - 4\mathbf{j})\) m s\(^{-1}\). Find

1. the acceleration of \(P\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\), [2]
2. the magnitude of \(\mathbf{F}\), [4]
3. the velocity of \(P\) at time \(t = 6\) s. [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]

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]