Vectors: exam review Exam review

A drone is at the point P and a landing pad is at the point Q. Relative to an origin O, the position vectors are \(\overrightarrow{OP} = 4\mathbf{i} - \mathbf{j} + 2\mathbf{k}\) and \(\overrightarrow{OQ} = \mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\), in metres. [6]

1. (a) Write down \(\overrightarrow{PQ}\) as a column vector. [2]
2. (b) Find the exact distance from P to Q. [2]
3. (c) Find the vector of magnitude 10 in the same direction as \(\overrightarrow{PQ}\). Give the components to 3 significant figures. [2]

In triangle OAB, \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\). The point M is the midpoint of OA. The point P lies on AB with AP:PB = 1:2. The point Q is such that OAQB is a parallelogram. [6]

1. (a) Find \(\overrightarrow{AB}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\). [1]
2. (b) Show that \(\overrightarrow{OP} = \dfrac{1}{3}(2\mathbf{a} + \mathbf{b})\). [2]
3. (c) Show that M, P and Q lie on a straight line. [3]

Let \(\mathbf{u} = \begin{pmatrix} 3 \\ p \\ -6 \end{pmatrix}\), \(\mathbf{v} = \begin{pmatrix} q \\ -10 \\ 4 \end{pmatrix}\) and \(\mathbf{w} = \begin{pmatrix} k \\ k + 2 \\ 2 \end{pmatrix}\), 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 \(|\mathbf{w}| = 6\), find the possible values of \(k\). Give your answers to 3 significant figures. [3]

The line \(l\) passes through the points A\((2, -1, 3)\) and B\((5, 1, -1)\). The point C has coordinates \((-1, -3, 7)\). [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 C lies on \(l\). [2]
3. (c) Write down parametric equations of \(l\). [2]

A ship is sailing in open water with constant velocity. At time \(t\) hours after noon its position vector is \(\mathbf{r} = \begin{pmatrix} -4 \\ 7 \end{pmatrix} + t\begin{pmatrix} 9 \\ -4 \end{pmatrix}\), in km relative to the lighthouse L. The first component is the distance east and the second is the distance north. [7]

1. (a) Find the speed of the ship. [2]
2. (b) Find the bearing on which the ship is moving. Give your answer to the nearest degree. [2]
3. (c) Find the distance of the ship from the lighthouse L when \(t = 2.5\). [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 \(|\mathbf{a}| = 4\) and \(|\mathbf{b}| = 7\). The angle between them is 140°. Find \(\mathbf{a} \cdot \mathbf{b}\). Give your answer to 3 significant figures. [2]
2. (b) Let \(\mathbf{p} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix}\). Find the angle between \(\mathbf{p}\) and \(\mathbf{q}\). Give your answer to the nearest degree. [3]
3. (c) The vector \(\mathbf{s} = 5\mathbf{i} + t\mathbf{j} - \mathbf{k}\) is perpendicular to \(\mathbf{z} = 3\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}\). 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 \(|\mathbf{a}| = 6\) and \(|\mathbf{b}| = 7\). The angle between them is 125°. Find \(|\mathbf{a} \times \mathbf{b}|\). Give your answer to 3 significant figures. [2]
2. (b) Let \(\mathbf{p} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}\) and \(\mathbf{q} = \mathbf{i} + 4\mathbf{j} - \mathbf{k}\). 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 model train runs on a straight track. Its velocity at time \(t\) seconds is \(v = 7\cos(0.8t)\) m s⁻¹. A second model train runs on a parallel straight track. When its displacement from the station is \(s\) metres, its velocity is \(v = 4 + 0.3s^2\) m s⁻¹. [6]

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

A drone flies in a horizontal plane. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} 3 t^{2} - 4 \\ 6 t - 5 \end{pmatrix}\) m s⁻¹, for \(t \geq 0\). When \(t = 0\) its position vector relative to a fixed origin O is \(\begin{pmatrix} 2 \\ -1 \end{pmatrix}\) metres. [7]

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

A stone is thrown from a point O on level ground. Its position vector relative to O at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix} 9t \\ 14t - 4.9t^2 \end{pmatrix}\) metres, where the first component is horizontal and the second is vertically upward. [6]

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

A bead slides on a circular wire. Its position vector at time \(t\) seconds, relative to a fixed origin O, is \(\mathbf{r} = \begin{pmatrix} 6 + 4\cos(0.5t) \\ -3 + 4\sin(0.5t) \end{pmatrix}\) metres. A second bead starts at the highest point of the circle and goes round clockwise, taking 8 seconds for each complete turn, on the same circle. [8]

1. (a) Show that the bead moves on a circle, and write down the centre and the radius of the circle. [3]
2. (b) State whether the bead 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 bead at time \(t\) seconds. [3]

Two straight cables in a stadium roof are modelled by lines. The lines \(l_1\) and \(l_2\) are given by \(l_1:\ \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda\begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}\) and \(l_2:\ \mathbf{r} = 4\mathbf{i} + 6\mathbf{j} + 3\mathbf{k} + \mu\left(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}\right)\). [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 \((8, 3, 7)\). 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 \(x = s + 2,\ y = 5 - s,\ z = 2 s - 1\). Show that \(l_1\) and \(l_3\) do not intersect. [3]

Two drones, A and B, fly in a light show. 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 \(t = 0\) A is at \((10, 20, 30)\) and moves with constant velocity \(\begin{pmatrix} 4 \\ -2 \\ 1 \end{pmatrix}\) m/s. At \(t = 0\) B is at \((-8, 32, 12)\) and moves with constant speed 6 m/s in the direction of \(\begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix}\). [14]

1. (a) Find the speed of A. Give your answer to 3 significant figures. [2]
2. (b) Show that the velocity of B is \(\begin{pmatrix} 2 \\ -4 \\ 4 \end{pmatrix}\). [3]
3. (c) Write down the position vector of B at time \(t\). [1]
4. (d) Find the distance between A and B when \(t = 3\). Give your answer to 3 significant figures. [2]
5. (e) Show that A and B 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 sail ABC is held in the air by three poles. The corners are A\((2, 1, -1)\), B\((5, 3, 1)\) and C\((4, 5, 2)\), in metres. [14]

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

C is a carriage on a circular track. Its position vector relative to the centre O of the track at time \(t\) seconds, where \(t \geq 0\), is \(\mathbf{r}_C = \begin{pmatrix} 8\cos(0.05t^2) \\ 8\sin(0.05t^2) \end{pmatrix}\) metres. A second carriage D starts from the same point as C, 5 seconds after C, and moves in exactly the same way. [12]

1. (a) Show that C 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 C at time \(t\). [3]
3. (c) Find the speed of C when \(t = 4\). [2]
4. (d) Find the number of complete revolutions that C makes in the first 20 seconds. [2]
5. (e) Write down an expression for the position vector \(\mathbf{r}_D\) of D at time \(t\), for \(t \geq 5\). [1]
6. (f) Find the distance between C and D when \(t = 9\). Give your answer to 3 significant figures. [2]

A robot moves along a straight rail. The velocity of the robot at time \(t\) seconds is \(v = t^{2} - 9 t + 14\) m s⁻¹, for \(t \geq 0\). [7]

1. (a) Find the times when the robot is at rest, and for each time say whether the robot changes direction. [3]
2. (b) Find the total distance travelled by the robot in the first 8 seconds. Give your answer to 3 significant figures. [2]
3. (c) The robot 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 ball is thrown from a point 1.5 m above level ground, with speed 16 m s⁻¹ at an angle of 35° 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 ball 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 ball 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 ball above the ground. [2]
4. (d) Find the speed of the ball when it lands, and the angle below the horizontal at which it lands. [3]
5. (e) The ball passes a fence, of height 5.0 m, at a horizontal distance 15 m from O. Decide whether the ball clears the fence, and find the vertical distance between the ball and the top of the fence. [2]