1.10d Vector operations: addition and scalar multiplication

\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]

At time \(t = 0\), a particle \(P\) of mass \(3\) kg is at rest at the point \(A\) with position vector \((j - 3k)\) m. Two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) then act on the particle \(P\) and it passes through the point \(B\) with position vector \((8i - 3j + 5k)\) m. Given that \(\mathbf{F}_1 = (4i - 2j + 5k)\) N and \(\mathbf{F}_2 = (8i - 4j + 7k)\) N and that \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are the only two forces acting on \(P\), find the velocity of \(P\) as it passes through \(B\), giving your answer as a vector. [7]

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]

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]

Two fixed points, \(A\) and \(B\), have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) relative to the origin \(O\), and a variable point \(P\) has position vector \(\mathbf{r}\).

1. Give a geometrical description of the locus of \(P\) when \(\mathbf{r}\) satisfies the equation \(\mathbf{r} = \lambda\mathbf{a}\), where \(0 \leq \lambda \leq 1\). [2]
2. Given that \(P\) is a point on the line \(AB\), use a property of the vector product to explain why \((\mathbf{r} - \mathbf{a}) \times (\mathbf{r} - \mathbf{b}) = \mathbf{0}\). [2]
3. Give a geometrical description of the locus of \(P\) when \(\mathbf{r}\) satisfies the equation \(\mathbf{r} \times (\mathbf{a} - \mathbf{b}) = \mathbf{0}\). [3]

Relative to a fixed origin \(O\), the position vectors of the points \(A\), \(B\) and \(C\) are $$\overrightarrow{OA} = -3\mathbf{i} + \mathbf{j} - 9\mathbf{k}, \quad \overrightarrow{OB} = \mathbf{i} - \mathbf{k}, \quad \overrightarrow{OC} = 5\mathbf{i} + 2\mathbf{j} - 5\mathbf{k} \text{ respectively}.$$

1. Find the cosine of angle \(ABC\). [4]

The line \(L\) is the angle bisector of angle \(ABC\).

1. Show that an equation of \(L\) is \(\mathbf{r} = \mathbf{i} - \mathbf{k} + t(\mathbf{i} + 2\mathbf{j} - 7\mathbf{k})\). [4]
2. Show that \(|\overrightarrow{AB}| = |\overrightarrow{AC}|\). [2]

The circle \(S\) lies inside triangle \(ABC\) and each side of the triangle is a tangent to \(S\).

1. Find the position vector of the centre of \(S\). [7]
2. Find the radius of \(S\). [5]

\(A\) and \(B\) are fixed points in the \(x\)-\(y\) plane. The position vectors of \(A\) and \(B\) are \(\mathbf{a}\) and \(\mathbf{b}\) respectively. State, with reference to points \(A\) and \(B\), the geometrical significance of

1. the quantity \(|\mathbf{a} - \mathbf{b}|\), [1]
2. the vector \(\frac{1}{2}(\mathbf{a} + \mathbf{b})\). [1]

The circle \(P\) is the set of points with position vector \(\mathbf{p}\) in the \(x\)-\(y\) plane which satisfy $$\left|\mathbf{p} - \frac{1}{2}(\mathbf{a} + \mathbf{b})\right| = \frac{1}{2}|\mathbf{a} - \mathbf{b}|.$$

1. State, in terms of \(\mathbf{a}\) and \(\mathbf{b}\),
   1. the position vector of the centre of \(P\), [1]
   2. the radius of \(P\). [1]

It is now given that \(\mathbf{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}\), \(\mathbf{b} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}\) and \(\mathbf{p} = \begin{pmatrix} x \\ y \end{pmatrix}\).

1. Find a cartesian equation of \(P\). [4]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle \(P\) is moving on a smooth horizontal surface under the action of a single force \(\mathbf{F}\) N. At time \(t\) seconds, where \(t \geq 0\), the velocity \(\mathbf{v} \mathrm{m s}^{-1}\) of \(P\), relative to a fixed origin \(O\), is given by $$\mathbf{v} = (1 - 2t)\mathbf{i} + (2t^2 + t - 13)\mathbf{j}.$$

1. Show that \(P\) is never stationary. [2]
2. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the acceleration of \(P\) at time \(t\). [1]

The mass of \(P\) is 0.5 kg.

1. Determine the magnitude of \(\mathbf{F}\) when \(P\) is moving in the direction of the vector \(-2\mathbf{i} + \mathbf{j}\). Give your answer correct to 3 significant figures. [5]

When \(t = 1\), \(P\) is at the point with position vector \(\frac{1}{6}\mathbf{j}\).

1. Determine the bearing of \(P\) from \(O\) at time \(t = 1.5\). [5]

A particle \(P\) of mass \(m \text{kg}\) is moving on a smooth horizontal surface under the action of two constant horizontal forces \((-4\mathbf{i} + 2\mathbf{j}) \text{N}\) and \((a\mathbf{i} + b\mathbf{j}) \text{N}\). The resultant of these two forces is \(\mathbf{R} \text{N}\). It is given that \(\mathbf{R}\) acts in a direction which is parallel to the vector \(-\mathbf{i} + 3\mathbf{j}\).

1. Show that \(3a + b = 10\). [3]

It is given that \(a = 6\) and that \(P\) moves with an acceleration of magnitude \(5\sqrt{10} \text{ms}^{-2}\).

1. Determine the value of \(m\). [4]

Two forces \(\begin{bmatrix}3\\-2\end{bmatrix}\) N and \(\begin{bmatrix}-7\\-5\end{bmatrix}\) N act on a particle. Find the resultant force. Circle your answer. [1 mark] \(\begin{bmatrix}-21\\10\end{bmatrix}\) N \(\begin{bmatrix}-4\\-7\end{bmatrix}\) N \(\begin{bmatrix}4\\3\end{bmatrix}\) N \(\begin{bmatrix}10\\7\end{bmatrix}\) N

Two points \(A\) and \(B\) lie in a horizontal plane and have coordinates \((-2, 7)\) and \((3, 19)\) respectively. A particle moves in a straight line from \(A\) to \(B\) under the action of a constant resultant force of magnitude 6.5 N Express the resultant force in vector form. [3 marks]

Two particles, \(P\) and \(Q\), are initially at rest at the same point on a horizontal plane. A force of \(\begin{bmatrix} 4 \\ 0 \end{bmatrix}\) N is applied to \(P\). A force of \(\begin{bmatrix} 8 \\ 15 \end{bmatrix}\) N is applied to \(Q\).

1. Calculate, to the nearest degree, the acute angle between the two forces. [2 marks]
2. The particles begin to move under the action of the respective forces. \(P\) and \(Q\) have the same mass. \(P\) has an acceleration of magnitude 5 m s\(^{-2}\) Find the magnitude of the acceleration of \(Q\). [3 marks]

A particle, \(P\), is initially at rest on a smooth horizontal surface. A resultant force of \(\begin{bmatrix} 12 \\ 9 \end{bmatrix}\) N is then applied to \(P\), so that it moves in a straight line.

1. Find the magnitude of the resultant force. [1 mark]
2. Two fixed points \(A\) and \(B\) have position vectors $$\overrightarrow{OA} = \begin{bmatrix} 3 \\ 7 \end{bmatrix} \text{ metres} \quad \text{and} \quad \overrightarrow{OB} = \begin{bmatrix} k \\ k-1 \end{bmatrix} \text{ metres}$$ with respect to a fixed origin, \(O\) \(P\) moves in a straight line parallel to \(\overrightarrow{AB}\)
   1. Find \(\overrightarrow{AB}\) in terms of \(k\) [1 mark]
   2. Find the value of \(k\) [2 marks]

Two forces, \(\mathbf{F}_1 = 3\mathbf{i} + 2\mathbf{j}\) newtons and \(\mathbf{F}_2 = \mathbf{i} - 3\mathbf{j}\) newtons, are added together to find a resultant force, \(\mathbf{R}\) newtons. This vector addition can be represented using a diagram. Identify the diagram below which correctly represents this vector addition. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_14}

A quadrilateral has vertices A, B, C and D with position vectors given by $$\overrightarrow{OA} = \begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix}, \overrightarrow{OB} = \begin{pmatrix} -1 \\ 2 \\ 7 \end{pmatrix}, \overrightarrow{OC} = \begin{pmatrix} 0 \\ 7 \\ 6 \end{pmatrix} \text{ and } \overrightarrow{OD} = \begin{pmatrix} 4 \\ 10 \\ 0 \end{pmatrix}$$

1. Write down the vector \(\overrightarrow{AB}\) [1 mark]
2. Show that ABCD is a parallelogram, but not a rhombus. [5 marks]

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]

At time \(t\) seconds a particle, \(P\), has position vector \(\mathbf{r}\) metres, with respect to a fixed origin, such that $$\mathbf{r} = (t^3 - 5t^2)\mathbf{i} + (8t - t^2)\mathbf{j}$$

1. Find the exact speed of \(P\) when \(t = 2\) [4 marks]
2. Bella claims that the magnitude of acceleration of \(P\) will never be zero. Determine whether Bella's claim is correct. Fully justify your answer. [3 marks]

Two forces, \(\mathbf{F_1}\) and \(\mathbf{F_2}\), are acting on a particle of mass 3 kilograms. It is given that $$\mathbf{F_1} = \begin{pmatrix} a \\ 23 \end{pmatrix} \text{ newtons and } \mathbf{F_2} = \begin{pmatrix} 4 \\ b \end{pmatrix} \text{ newtons}$$ where \(a\) and \(b\) are constants. The particle has an acceleration of \(\begin{pmatrix} 4b \\ a \end{pmatrix}\) m s\(^{-2}\) Find the value of \(a\) and the value of \(b\) [4 marks]

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 \(3\mathbf{i} - 7\mathbf{j}\) and the point \(B\) has position vector \(8\mathbf{i} + 3\mathbf{j}\).

1. find the vector \(\overrightarrow{AB}\) [2]
2. Find \(|\overrightarrow{AB}|\). Give your answer as a simplified surd. [2]

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]