1.10a Vectors in 2D: i,j notation and column vectors

A cyclist \(P\) is cycling due north at a constant speed of 20 km h\(^{-1}\). At 12 noon another cyclist \(Q\) is due west of \(P\). The speed of \(Q\) is constant at 10 km h\(^{-1}\). Find the course which \(Q\) should set in order to pass as close to \(P\) as possible, giving your answer as a bearing. [5]

\includegraphics{figure_2} Boat \(A\) is travelling with constant speed 7.9 m s\(^{-1}\) on a course with bearing 035°. Boat \(B\) is travelling with constant speed 10.5 m s\(^{-1}\) on a course with bearing 330°. At one instant, the boats are 1500 m apart with \(B\) on a bearing of 125° from \(A\) (see diagram).

1. Find the magnitude and the bearing of the velocity of \(B\) relative to \(A\). [5]
2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion. [2]
3. Find the time taken from the instant when \(A\) and \(B\) are 1500 m apart to the instant when \(A\) and \(B\) are at the point of closest approach. [2]

A particle \(P\) moves in a plane such that its position vector \(\mathbf{r}\) metres at time \(t\) seconds \((t > 0)\) satisfies the differential equation $$\frac{d\mathbf{r}}{dt} - \frac{2}{t}\mathbf{r} = 4i$$ When \(t = 1\), the particle is at the point with position vector \((i + j)\) m. Find \(\mathbf{r}\) in terms of \(t\). [9]

The points \(P\) and \(Q\) have position vectors \(4i - 6j - 12k\) and \(2i + 4j + 4k\) respectively, relative to a fixed origin \(O\). Three forces, \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\), act along \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\) and \(\overrightarrow{QP}\) respectively, and have magnitudes \(7\) N, \(3\) N and \(3\sqrt{10}\) N respectively.

1. Express \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) in vector form. [3]

1. Show that the resultant of \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) is \((2i - 10j - 16k)\) N. [2]

1. Find a vector equation of the line of action of this resultant, giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors and \(\lambda\) is a parameter. [5]

A particle moves in a plane in such a way that its position vector \(\mathbf{r}\) metres at time \(t\) seconds satisfies the differential equation $$\frac{d^2\mathbf{r}}{dt^2} - 2\frac{d\mathbf{r}}{dt} = 0$$ When \(t = 0\), the particle is at the origin and is moving with velocity \((4i + 2j)\) m s\(^{-1}\). Find \(\mathbf{r}\) in terms of \(t\). [7]

Three forces \(\mathbf{F}_1 = (3i - j + k)\) N, \(\mathbf{F}_2 = (2i - k)\) N, and \(\mathbf{F}_3\) act on a rigid body. The force \(\mathbf{F}_1\) acts through the point with position vector \((i + 2j + k)\) m, the force \(\mathbf{F}_2\) acts through the point with position vector \((i - 2j)\) m and the force \(\mathbf{F}_3\) acts through the point with position vector \((i + j + k)\) m. Given that the system \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) reduces to a couple \(\mathbf{G}\),

1. find \(\mathbf{G}\). [6]

The line of action of \(\mathbf{F}_3\) is changed so that the system \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) now reduces to a couple \((6i + 8j + 2k)\) N m.

1. Find an equation of the new line of action of \(\mathbf{F}_3\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors. [5]

A particle of mass 0.5 kg is at rest at the point with position vector \((2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})\) m. The particle is then acted upon by two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\). These are the only two forces acting on the particle. Subsequently, the particle passes through the point with position vector \((4\mathbf{i} + 5\mathbf{j} - 5\mathbf{k})\) m with speed 12 m s\(^{-1}\). Given that \(\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j} - \mathbf{k})\) N, find \(\mathbf{F}_2\). [9]

A particle \(P\) moves in the \(x\)-\(y\) plane and has position vector \(\mathbf{r}\) metres at time \(t\) seconds. It is given that \(\mathbf{r}\) satisfies the differential equation $$\frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2} = 2\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}.$$ When \(t = 0\), \(P\) is at the point with position vector \(3\mathbf{i}\) metres and is moving with velocity \(\mathbf{j}\) m s\(^{-1}\).

1. Find \(\mathbf{r}\) in terms of \(t\). [8]
2. Describe the path of \(P\), giving its cartesian equation. [2]

A force system consists of three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) acting on a rigid body. \(\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j})\) N and acts at the point with position vector \((-\mathbf{i} + 4\mathbf{j})\) m. \(\mathbf{F}_2 = (-\mathbf{j} + \mathbf{k})\) N and acts at the point with position vector \((2\mathbf{i} + \mathbf{j} + \mathbf{k})\) m. \(\mathbf{F}_3 = (3\mathbf{i} - \mathbf{j} + \mathbf{k})\) N and acts at the point with position vector \((\mathbf{i} - \mathbf{j} + 2\mathbf{k})\) m. It is given that this system can be reduced to a single force \(\mathbf{R}\).

1. Find \(\mathbf{R}\). [2]
2. Find a vector equation of the line of action of \(\mathbf{R}\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors and \(\lambda\) is a parameter. [10]

A particle moves from the point \(A\) with position vector \((3\mathbf{i} - \mathbf{j} + 3\mathbf{k})\) m to the point \(B\) with position vector \((\mathbf{i} - 2\mathbf{j} - 4\mathbf{k})\) m under the action of the force \((2\mathbf{i} - 3\mathbf{j} - \mathbf{k})\) N. Find the work done by the force. [4]

A particle \(P\) moves in the \(x\)-\(y\) plane so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds satisfies the differential equation $$\frac{d^2\mathbf{r}}{dt^2} - 4\mathbf{r} = -3e^t\mathbf{j}$$ When \(t = 0\), the particle is at the origin and is moving with velocity \((2\mathbf{i} + \mathbf{j})\) ms\(^{-1}\). Find \(\mathbf{r}\) in terms of \(t\). [10]

Two forces \(\mathbf{F}_1 = (3\mathbf{i} + \mathbf{k})\) N and \(\mathbf{F}_2 = (4\mathbf{i} + \mathbf{j} - \mathbf{k})\) N act on a rigid body. The force \(\mathbf{F}_1\) acts at the point with position vector \((2\mathbf{i} - \mathbf{j} + 3\mathbf{k})\) m and the force \(\mathbf{F}_2\) acts at the point with position vector \((-3\mathbf{i} + 2\mathbf{k})\) m. The two forces are equivalent to a single force \(\mathbf{R}\) acting at the point with position vector \((\mathbf{i} + 2\mathbf{j} + \mathbf{k})\) m together with a couple of moment \(\mathbf{G}\). Find,

1. \(\mathbf{R}\), [2]
2. \(\mathbf{G}\). [4]

A third force \(\mathbf{F}_3\) is now added to the system. The force \(\mathbf{F}_3\) acts at the point with position vector \((2\mathbf{i} - \mathbf{k})\) m and the three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are equivalent to a couple.

1. Find the magnitude of the couple. [6]

A particle \(P\) moves in a plane such that its position vector \(\mathbf{r}\) metres at time \(t\) seconds \((t > 0)\) satisfies the differential equation $$\frac{d\mathbf{r}}{dt} - \frac{2}{t}\mathbf{r} = 4\mathbf{i}$$ When \(t = 1\), the particle is at the point with position vector \((\mathbf{i} + \mathbf{j})\) m. Find \(\mathbf{r}\) in terms of \(t\). [9]

The points \(P\) and \(Q\) have position vectors \(4\mathbf{i} - 6\mathbf{j} - 12\mathbf{k}\) and \(2\mathbf{i} + 4\mathbf{j} + 4\mathbf{k}\) respectively, relative to a fixed origin \(O\). Three forces, \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\), act along \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\) and \(\overrightarrow{QP}\) respectively, and have magnitudes 7 N, 3 N and \(3\sqrt{10}\) N respectively.

1. Express \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) in vector form. [3]
2. Show that the resultant of \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) is \((2\mathbf{i} - 10\mathbf{j} - 16\mathbf{k})\) N. [2]
3. Find a vector equation of the line of action of this resultant, giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors and \(\lambda\) is a parameter. [5]

Four buoys on the surface of a large, calm lake are located at \(A\), \(B\), \(C\) and \(D\) with position vectors given by $$\overrightarrow{OA} = \begin{bmatrix} 410 \\ 710 \end{bmatrix}, \overrightarrow{OB} = \begin{bmatrix} -210 \\ 530 \end{bmatrix}, \overrightarrow{OC} = \begin{bmatrix} -340 \\ -310 \end{bmatrix} \text{ and } \overrightarrow{OD} = \begin{bmatrix} 590 \\ -40 \end{bmatrix}$$ All values are in metres.

1. Prove that the quadrilateral \(ABCD\) is a trapezium but not a parallelogram. [5 marks]
2. A speed boat travels directly from \(B\) to \(C\) at a constant speed in 50 seconds. Find the speed of the boat between \(B\) and \(C\). [4 marks]

A number of forces act on a particle such that the resultant force is \(\begin{pmatrix} 6 \\ -3 \end{pmatrix}\) N One of the forces acting on the particle is \(\begin{pmatrix} 8 \\ -5 \end{pmatrix}\) N Calculate the total of the other forces acting on the particle. Circle your answer. \(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\) N \quad \(\begin{pmatrix} 14 \\ -8 \end{pmatrix}\) N \quad \(\begin{pmatrix} -2 \\ 2 \end{pmatrix}\) N \quad \(\begin{pmatrix} -14 \\ 8 \end{pmatrix}\) N [1 mark]

A particle, \(P\), is moving with constant velocity \(8\mathbf{i} - 12\mathbf{j}\) A second particle, \(Q\), is moving with constant velocity \(a\mathbf{i} + 9\mathbf{j}\) \(Q\) travels in a direction which is parallel to the motion of \(P\). Find \(a\). Circle your answer. \(-6\) \quad \(-5\) \quad \(5\) \quad \(6\) [1 mark]

Two particles \(P\) and \(Q\) are moving in separate straight lines across a smooth horizontal surface. \(P\) moves with constant velocity \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) \(Q\) moves from position vector \((5\mathbf{i} - 7\mathbf{j})\) metres to position vector \((14\mathbf{i} + 5\mathbf{j})\) metres during a 3 second period.

1. Show that \(P\) and \(Q\) move along parallel lines. [3 marks]
2. Stevie says Q is also moving with a constant velocity of \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) Explain why Stevie may be incorrect. [1 mark]
3. A third particle \(R\) is moving with a constant speed of 4 m s\(^{-1}\), in a straight line, across the same surface. \(P\) and \(R\) move along lines that intersect at a fixed point \(X\) It is given that: • \(P\) passes through \(X\) exactly 2 seconds after \(R\) passes through \(X\) • \(P\) and \(R\) are exactly 13 metres apart 3 seconds after \(R\) passes through \(X\) Show that \(P\) and \(R\) move along perpendicular lines. [5 marks]

The three forces \(\mathbf{F_1}\), \(\mathbf{F_2}\) and \(\mathbf{F_3}\) are acting on a particle. \(\mathbf{F_1} = (25\mathbf{i} + 12\mathbf{j})\) N \(\mathbf{F_2} = (-7\mathbf{i} + 5\mathbf{j})\) N \(\mathbf{F_3} = (15\mathbf{i} - 28\mathbf{j})\) N The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. The resultant of these three forces is \(\mathbf{F}\) newtons.

The fourth force, \(\mathbf{F_4}\), is applied to the particle so that the four forces are in equilibrium. Find \(\mathbf{F_4}\), giving your answer in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [1 mark]

Given that the point \(A\) has position vector \(x\mathbf{i} - \mathbf{j}\), the point \(B\) has position vector \(-2\mathbf{i} + y\mathbf{j}\) and \(\overrightarrow{AB} = -3\mathbf{i} + 4\mathbf{j}\), find

1. the values of \(x\) and \(y\) [3]
2. a unit vector in the direction of \(\overrightarrow{AB}\). [2]

\(OABC\) is a parallelogram with \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OC} = \mathbf{c}\). \(P\) is the midpoint of \(AC\). \includegraphics{figure_7}

1. Find the following in terms of \(\mathbf{a}\) and \(\mathbf{c}\), simplifying your answers.
   1. \(\overrightarrow{AC}\) [1]
   2. \(\overrightarrow{OP}\) [2]
2. Hence prove that the diagonals of a parallelogram bisect one another. [4]

A vector is given by \(\mathbf{a} = \begin{bmatrix} 2 \\ -1 \\ -3 \end{bmatrix}\) Which vector is not perpendicular to \(\mathbf{a}\)? Circle your answer. \(\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix}\) \quad \(\begin{bmatrix} 5 \\ -1 \\ 3 \end{bmatrix}\) \quad \(\begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}\) [1 mark]

The vectors \(\mathbf{a}\) and \(\mathbf{b}\) are defined by \(\mathbf{a} = 2\mathbf{i} - \mathbf{j}\) and \(\mathbf{b} = \mathbf{i} - 3\mathbf{j}\).

1. Find a unit vector in the direction of \(\mathbf{a}\). [2]
2. Determine the angle \(\mathbf{b}\) makes with the \(x\)-axis. [2]
3. The vector \(\mu\mathbf{a} + \mathbf{b}\) is parallel to \(4\mathbf{i} - 5\mathbf{j}\).
   1. Find the vector \(\mu\mathbf{a} + \mathbf{b}\) in terms of \(\mu\), \(\mathbf{i}\) and \(\mathbf{j}\). [1]
   2. Determine the value of \(\mu\). [4]