Vectors: exam review Exam review: second paper

A hiker’s beacon is at the point P and a rescue helicopter is at the point Q. Relative to an origin O, the position vectors are \(\overrightarrow{OP} = -2\mathbf{i} + 5\mathbf{j} + \mathbf{k}\) and \(\overrightarrow{OQ} = 3\mathbf{i} + \mathbf{j} - 3\mathbf{k}\), in metres. [6]

1. (a) Write down \(\overrightarrow{PQ}\) as a column vector. [2]
2. (b) Find the exact distance between P and Q. [2]
3. (c) Find the vector of magnitude 12 in the same direction as \(\overrightarrow{PQ}\). Give the components to 3 significant figures. [2]

In triangle OCD, \(\overrightarrow{OC} = \mathbf{c}\) and \(\overrightarrow{OD} = \mathbf{d}\). The point N is the midpoint of OD. The point R lies on CD with CR:RD = 2:1. The point S is such that OCSD is a parallelogram. [6]

1. (a) Find \(\overrightarrow{CD}\) in terms of \(\mathbf{c}\) and \(\mathbf{d}\). [1]
2. (b) Show that \(\overrightarrow{OR} = \dfrac{1}{3}(\mathbf{c} + 2\mathbf{d})\). [2]
3. (c) Show that N, R and S lie on a straight line. [3]

Let \(\mathbf{u} = \begin{pmatrix} p \\ 9 \\ -6 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} -2 \\ q \\ 4 \end{pmatrix}\) and \(\mathbf{w} = \begin{pmatrix} k + 1 \\ 2k \\ 3 \end{pmatrix}\), where \(p\), \(q\) and \(k\) are constants. [6]

1. (a) Given that \(\mathbf{u}\) is parallel to \(\mathbf{v}\), find the value of \(p\) and the value of \(q\). [3]
2. (b) Given that \(|\mathbf{w}| = 8\), find the possible values of \(k\). Give your answers to 3 significant figures. [3]

The line \(l\) passes through the points P\((-3, 4, 1)\) and Q\((1, 2, 6)\). The point R has coordinates \((5, 0, 10)\). [6]

1. (a) Find a vector equation of \(l\). Give your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\). [2]
2. (b) Determine whether R lies on \(l\). [2]
3. (c) Write down parametric equations of \(l\). [2]

A yacht is sailing in a race with constant velocity. At time \(t\) hours after the start its position vector is \(\mathbf{r} = \begin{pmatrix} 6 \\ -2 \end{pmatrix} + t\begin{pmatrix} -7 \\ 5 \end{pmatrix}\), in km relative to the race marker M. The first component is the distance east and the second is the distance north. [7]

1. (a) Find the speed of the yacht. [2]
2. (b) Find the bearing on which the yacht is moving. Give your answer to the nearest degree. [2]
3. (c) Find the distance of the yacht from the race marker M when \(t = 3.5\). [3]

The scalar product (dot product) of two vectors is written \(\mathbf{a} \cdot \mathbf{b}\). [7]

1. (a) The vectors \(\mathbf{a}\) and \(\mathbf{b}\) have \(|\mathbf{a}| = 5\) and \(|\mathbf{b}| = 9\). The angle between them is 110°. Find \(\mathbf{a} \cdot \mathbf{b}\). Give your answer to 3 significant figures. [2]
2. (b) Let \(\mathbf{p} = \begin{pmatrix} 4 \\ 1 \\ -2 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} -3 \\ 5 \\ 2 \end{pmatrix}\). Find the angle between \(\mathbf{p}\) and \(\mathbf{q}\). Give your answer to the nearest degree. [3]
3. (c) The vector \(\mathbf{s} = t\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}\) is perpendicular to \(\mathbf{z} = 2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}\). Find the value of \(t\). [2]

The vector product (cross product) of two vectors is written \(\mathbf{a} \times \mathbf{b}\). [7]

1. (a) The vectors \(\mathbf{a}\) and \(\mathbf{b}\) have \(|\mathbf{a}| = 5\) and \(|\mathbf{b}| = 8\). The angle between them is 115°. Find \(|\mathbf{a} \times \mathbf{b}|\). Give your answer to 3 significant figures. [2]
2. (b) Let \(\mathbf{p} = -\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\) and \(\mathbf{q} = 3\mathbf{i} + \mathbf{j} - 2\mathbf{k}\). Find \(\mathbf{p} \times \mathbf{q}\). [2]
3. (c) Find the component of \(\mathbf{p}\) in the direction of \(\mathbf{q}\). Give your answer to 3 significant figures. [3]

A toy tram runs on a straight track. Its velocity at time \(t\) seconds is \(v = 9\sin(0.6t)\) m s⁻¹. A second toy tram runs on a parallel straight track. When its displacement from the station is \(s\) metres, its velocity is \(v = 3 + 0.2s^2\) m s⁻¹. [6]

1. (a) Find the acceleration of the first tram when \(t = 2\). Give your answer to 3 significant figures. [2]
2. (b) Find the change in displacement of the first tram between \(t = 0\) and \(t = 4\). Give your answer to 3 significant figures. [2]
3. (c) Find the acceleration of the second tram when \(s = 3\). [2]

A buggy moves across a flat field. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} 6 t^{2} - 2 \\ 4 t - 7 \end{pmatrix}\) m s⁻¹, for \(t \geq 0\). When \(t = 0\) its position vector relative to a fixed origin O is \(\begin{pmatrix} -3 \\ 4 \end{pmatrix}\) metres. [7]

1. (a) Find the acceleration of the buggy when \(t = 3\). [2]
2. (b) Find an expression for the position vector of the buggy at time \(t\). [3]
3. (c) Find the speed of the buggy when \(t = 2\). Give your answer to 3 significant figures. [2]

A football is kicked from a point O on level ground. Its position vector relative to O at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix} 11t \\ 12t - 4.9t^2 \end{pmatrix}\) metres, where the first component is horizontal and the second is vertically upward. [6]

1. (a) Find the velocity vector of the football at time \(t\). [2]
2. (b) Find the greatest height of the football above the ground. Give your answer to 3 significant figures. [2]
3. (c) Find the horizontal distance from O to the point where the football lands. Give your answer to 3 significant figures. [2]

A carriage runs round a circular track. Its position vector at time \(t\) seconds, relative to a fixed origin O, is \(\mathbf{r} = \begin{pmatrix} -2 + 3\sin(1.5t) \\ 5 + 3\cos(1.5t) \end{pmatrix}\) metres. A second carriage starts at the lowest point of the circle and goes round anticlockwise, taking 5 seconds for each complete turn, on the same circle. [8]

1. (a) Show that the carriage moves on a circle, and write down the centre and the radius of the circle. [3]
2. (b) State whether the carriage goes round the circle clockwise or anticlockwise, and find the time it takes to go round once. Give your answer to 3 significant figures. [2]
3. (c) Write down an expression for the position vector of the second carriage at time \(t\) seconds. [3]

Two straight pipes in a tunnel are modelled by lines. The lines \(l_1\) and \(l_2\) are given by \(l_1:\ \mathbf{r} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix} + \lambda\begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}\) and \(l_2:\ \mathbf{r} = -2\mathbf{i} + 8\mathbf{j} - 5\mathbf{k} + \mu\left(2\mathbf{i} - \mathbf{j} + 2\mathbf{k}\right)\). [14]

1. (a) Show that \(l_1\) and \(l_2\) intersect, and find the coordinates of the point of intersection. [4]
2. (b) Find the acute angle between \(l_1\) and \(l_2\). [3]
3. (c) The point P has coordinates \((3, 6, -2)\). Find the shortest distance from P to \(l_2\). Give your answer to 3 significant figures. [4]
4. (d) A third line \(l_3\) has parametric equations \(x = 2 s + 1,\ y = s,\ z = 6 - s\). Show that \(l_1\) and \(l_3\) do not intersect. [3]

Two search helicopters, H and K, fly over a forest. Positions are measured in metres from a fixed point O, with \(\mathbf{i}\) east, \(\mathbf{j}\) north and \(\mathbf{k}\) vertically upwards, and \(t\) is the time in seconds. At \(t = 0\) H is at \((-50, 30, 20)\) and moves with constant velocity \(\begin{pmatrix} 10 \\ -5 \\ 3 \end{pmatrix}\) m/s. At \(t = 0\) K is at \((-80, 60, 25)\) and moves with constant speed 14 m/s in the direction of \(\begin{pmatrix} 6 \\ -3 \\ 2 \end{pmatrix}\). [14]

1. (a) Find the speed of H. Give your answer to 3 significant figures. [2]
2. (b) Show that the velocity of K is \(\begin{pmatrix} 12 \\ -6 \\ 4 \end{pmatrix}\). [3]
3. (c) Write down the position vector of K at time \(t\). [1]
4. (d) Find the distance between H and K when \(t = 5\). Give your answer to 3 significant figures. [2]
5. (e) Show that H and K do not collide. [3]
6. (f) Use your calculator to find the time at which they are closest together, and the distance between them at that time. Give your answers to 3 significant figures. [3]

A triangular solar panel DEF is held up by three posts. The corners are D\((1, -2, 4)\), E\((4, 0, 3)\) and F\((2, 3, 7)\), in metres. [14]

1. (a) Find \(\overrightarrow{DE}\) and \(\overrightarrow{DF}\) as column vectors. [2]
2. (b) Find the angle EDF. Give your answer to the nearest degree. [3]
3. (c) Find \(\overrightarrow{DE} \times \overrightarrow{DF}\). [2]
4. (d) Hence find the area of triangle DEF. Give your answer to 3 significant figures. [2]
5. (e) Find a unit vector that is perpendicular to both \(\overrightarrow{DE}\) and \(\overrightarrow{DF}\). [2]
6. (f) Find the shortest distance from F to the line DE. Give your answer to 3 significant figures. [3]

P is a cabin on a circular track. Its position vector relative to the centre O of the track at time \(t\) seconds, where \(t \geq 0\), is \(\mathbf{r}_P = \begin{pmatrix} 6\cos(0.08t^2) \\ 6\sin(0.08t^2) \end{pmatrix}\) metres. A second cabin Q starts from the same point as P, 4 seconds after P, and moves in exactly the same way. [12]

1. (a) Show that P moves on a circle with centre O, and write down the radius of the circle. [2]
2. (b) Find an expression for the velocity of P at time \(t\). [3]
3. (c) Find the speed of P when \(t = 3\). [2]
4. (d) Find the number of complete revolutions that P makes in the first 15 seconds. [2]
5. (e) Write down an expression for the position vector \(\mathbf{r}_Q\) of Q at time \(t\), for \(t \geq 4\). [1]
6. (f) Find the distance between P and Q when \(t = 8\). Give your answer to 3 significant figures. [2]

A cyclist rides along a straight road. The velocity of the cyclist at time \(t\) seconds is \(v = t^{2} - 10 t + 16\) m s⁻¹, for \(t \geq 0\). [7]

1. (a) Find the times when the cyclist is at rest, and for each time say whether the cyclist changes direction. [3]
2. (b) Find the total distance travelled by the cyclist in the first 10 seconds. Give your answer to 3 significant figures. [2]
3. (c) The cyclist starts at the point where \(s = 0\). Find the first time after it starts at which it returns to this point. Give your answer to 3 significant figures. [2]

A stone is thrown from a point 1.8 m above level ground, with speed 14 m s⁻¹ at an angle of 40° above the horizontal. Take O as the point on the ground directly below the point of release, with \(\mathbf{i}\) horizontal and \(\mathbf{j}\) vertically upward. Ignore air resistance and take \(g = 9.8\) m s⁻². [12]

1. (a) Write down the position vector of the stone at time \(t\) seconds after it is thrown, in the form \(\mathbf{r} = \begin{pmatrix} x \\ y \end{pmatrix}\) where \(x\) and \(y\) are in terms of \(t\). [2]
2. (b) Find the time that the stone takes to reach the ground, and the horizontal distance from O to the point where it lands. [3]
3. (c) Find the greatest height of the stone above the ground. [2]
4. (d) Find the speed of the stone when it lands, and the angle below the horizontal at which it lands. [3]
5. (e) The stone passes a wall, of height 6.5 m, at a horizontal distance 10 m from O. Decide whether the stone clears the wall, and find the vertical distance between the stone and the top of the wall. [2]