1.10e Position vectors: and displacement

Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \((4\mathbf{i} - 5\mathbf{j})\) m and \((12\mathbf{i} + \mathbf{j})\) m respectively, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors.

1. Find the distance \(XY\). [2 marks]

A particle \(P\) of mass \(2\) kg moves from \(X\) to \(Y\) in \(4\) seconds, in a straight line at a constant speed.

1. Show that the velocity vector of \(P\) is \((2\mathbf{i} + 1.5\mathbf{j}) \text{ ms}^{-1}\). [3 marks]

The particle continues beyond \(Y\) with the same constant velocity.

1. Write down an expression for the position vector of \(P\) \(t\) seconds after leaving \(X\). [2 marks]
2. Find the value of \(t\) when \(P\) is at the point with position vector \((16\mathbf{i} + 4\mathbf{j})\) m. [2 marks]

When it is moving with the same constant speed, \(P\) collides directly with another particle \(Q\), of mass \(4\) kg, which is at rest. \(P\) and \(Q\) coalesce and move together as a single particle.

1. Find the velocity vector of the combined particle after the collision. [5 marks]

\(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. The point \(A\) has position vector \(6\mathbf{j}\) m relative to an origin \(O\). At time \(t = 0\) a particle \(P\) starts from \(O\) and moves with constant velocity \((5\mathbf{i} + 2\mathbf{j})\) ms\(^{-1}\). At the same instant a particle \(Q\) starts from \(A\) and moves with constant velocity \(4\mathbf{i}\) ms\(^{-1}\).

1. Write down the position vectors of \(P\) and of \(Q\) at time \(t\) seconds. [3 marks]
2. Show that the distance \(d\) m between \(P\) and \(Q\) at time \(t\) seconds is such that $$d^2 = 5t^2 - 24t + 36.$$ [5 marks]
3. Find the value of \(t\) for which \(d^2\) is a minimum. [3 marks]
4. Hence find the minimum distance between \(P\) and \(Q\), and state the position vector of each particle when they are closest together. [4 marks]

Two ramblers, Alison and Bill, are out walking. At midday, Alison is at the fixed origin \(O\), and Bill is at the point with position vector \((-5\mathbf{i} + 12\mathbf{j})\) km relative to \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular, horizontal unit vectors. They are both walking with constant velocity – Alison at \((2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\), and Bill at a speed of \(2\sqrt{10}\) km h\(^{-1}\) in a direction parallel to the vector \((3\mathbf{i} + \mathbf{j})\).

1. Find the distance between the two ramblers at midday. [2 marks]
2. Show that the velocity of Bill is \((6\mathbf{i} + 2\mathbf{j})\) km h\(^{-1}\). [3 marks]
3. Show that, at time \(t\) hours after midday, the position vector of Bill relative to Alison is $$[(4t - 5)\mathbf{i} + (12 - 3t)\mathbf{j}] \text{ km.}$$ [5 marks]
4. Show that the distance, \(d\) km, between the two ramblers is given by $$d^2 = 25t^2 - 112t + 169.$$ [2 marks]
5. Using your answer to part \((d)\), find the length of time to the nearest minute for which the distance between the Alison and Bill is less than 11 km. [5 marks]

During a cricket match, the batsman hits the ball and begins running with constant velocity \(4\mathbf{i}\) m s\(^{-1}\) to try and score a run. When the batsman is at the fixed origin \(O\), the ball is thrown by a member of the opposing team with velocity \((^-8\mathbf{i} + 24\mathbf{j})\) m s\(^{-1}\) from the point with position vector \((30\mathbf{i} - 60\mathbf{j})\) m, where \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal perpendicular unit vectors. At time \(t\) seconds after the ball is thrown, the position vectors of the batsman and the ball are \(\mathbf{r}\) metres and \(\mathbf{s}\) metres respectively. In a model of the situation, the ball is assumed to travel horizontally and air resistance is considered to be negligible.

1. Find expressions for \(\mathbf{r}\) and \(\mathbf{s}\) in terms of \(t\). [3 marks]
2. Show that the ball hits the batsman and find the position vector of the batsman when this occurs. [5 marks]
3. Write down two reasons why the assumptions used in these calculations are unlikely to provide a realistic model. [2 marks]

A quad-bike, a truck and a car are moving on a large, open, horizontal surface in a desert plain. Relative to the quad-bike, which is travelling due west at its maximum speed of \(10 \text{ m s}^{-1}\), the truck is moving on a bearing of \(340°\). Relative to the car, which is travelling due east at a speed of \(15 \text{ m s}^{-1}\), the truck is moving on a bearing of \(300°\).

1. Show that the speed of the truck is approximately \(24.7 \text{ m s}^{-1}\) and that it is moving on a bearing of \(318°\), correct to the nearest degree. [8 marks]
2. At the instant when the truck is at a distance of \(400\) metres from the quad-bike, the bearing of the truck from the quad-bike is \(060°\). The truck continues to move with the same velocity as in part (a). The quad-bike continues to move at a speed of \(10 \text{ m s}^{-1}\). Find the bearing, to the nearest degree, on which the quad-bike should travel in order to approach the truck as closely as possible. [5 marks]

\(A\) 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]

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 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]

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]

\(OABC\) is a parallelogram with \(O\) as origin. \includegraphics{figure_6} The position vector of \(A\) is \(\mathbf{a}\) and the position vector of \(C\) is \(\mathbf{c}\). The midpoint of \(AB\) is \(D\). The point \(E\) divides the line \(CB\) such that \(CE : EB = 2 : 1\).

1. Find, in terms of \(\mathbf{a}\) and \(\mathbf{c}\),
   1. the vector \(\overrightarrow{AC}\),
   2. the position vector of \(D\),
   3. the position vector of \(E\). [3]
2. Determine whether or not \(\overrightarrow{DE}\) is parallel to \(\overrightarrow{AC}\), clearly stating your reason. [2]

The position vectors of the points A and B, relative to a fixed origin O, are given by $$\mathbf{a} = 4\mathbf{i} + 7\mathbf{j}, \quad\quad \mathbf{b} = \mathbf{i} + 3\mathbf{j},$$ respectively.

1. Find the vector \(\overrightarrow{AB}\). [2]
2. Determine the distance between the points A and B. [2]
3. The position vector of the point C is given by \(\mathbf{c} = -2\mathbf{i} + 5\mathbf{j}\). The point D is such that the distance between C and D is equal to the distance between A and B, and \(\overrightarrow{CD}\) is parallel to \(\overrightarrow{AB}\). Find the possible position vectors of the point D. [4]

At time \(t = 0\) seconds, a particle \(A\) has position vector \((6\mathbf{i} + 2\mathbf{j} - 8\mathbf{k})\) m relative to a fixed origin \(O\) and is moving with constant velocity \((3\mathbf{i} - \mathbf{j} + 4\mathbf{k})\) ms\(^{-1}\).

1. Write down the position vector of particle \(A\) at time \(t\) seconds and hence find the distance \(OA\) when \(t = 5\). [4]
2. The position vector, \(\mathbf{r}_B\) metres, of another particle \(B\) at time \(t\) seconds is given by $$\mathbf{r}_B = 3\sin\left(\frac{t}{2}\right)\mathbf{i} - 3\cos\left(\frac{t}{2}\right)\mathbf{j} + 5\mathbf{k}.$$
   1. Show that \(B\) is moving with constant speed.
   2. Determine the smallest value of \(t\) such that particles \(A\) and \(B\) are moving perpendicular to each other. [7]

Relative to a fixed origin \(O\),

• the point \(A\) has position vector \(-2\mathbf{i} + 3\mathbf{j}\),
• the point \(B\) has position vector \(3\mathbf{i} + p\mathbf{j}\), where \(p\) is constant,

Given that \(|\overrightarrow{AB}| = 5\sqrt{2}\), find the possible values for \(p\). [3]

Relative to a fixed origin \(O\), • the point \(A\) has position vector \(\mathbf{5i + 3j - 2k}\) • the point \(B\) has position vector \(\mathbf{7i + j + 2k}\) • the point \(C\) has position vector \(\mathbf{4i + 8j - 3k}\)

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

Given that \(ABCD\) is a parallelogram,

1. find the position vector of the point \(D\). [2]

The point \(E\) is positioned such that • \(ACE\) is a straight line • \(AC:CE = 2:1\)

1. Find the coordinates of the point \(E\). [2]

Points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\). Point \(C\) lies on \(AB\) such that \(AC : CB = p : 1\).

1. Show that the position vector of \(C\) is \(\frac{1}{p+1}(\mathbf{a} + p\mathbf{b})\). [3]

It is now given that \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) and \(\mathbf{b} = -6\mathbf{i} + 4\mathbf{j} + 12\mathbf{k}\), and that \(C\) lies on the \(y\)-axis.

1. Find the value of \(p\). [4]
2. Write down the position vector of \(C\). [1]

The points A, B and C have position vectors \(\mathbf{3i - 4j + 2k}\), \(\mathbf{-i + 6k}\) and \(\mathbf{7i - 4j - 2k}\) respectively. M is the midpoint of BC.

1. Show that the magnitude of \(\overrightarrow{OM}\) is equal to \(\sqrt{17}\). [2]

Point D is such that \(\overrightarrow{BC} = \overrightarrow{AD}\).

1. Show that position vector of the point D is \(\mathbf{1i - 8j - 6k}\). [3]

The points \(A\), \(B\) and \(C\) have position vectors \(\begin{pmatrix} -2 \\ 1 \end{pmatrix}\), \(\begin{pmatrix} 2 \\ 5 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 3 \end{pmatrix}\) respectively. \(M\) is the midpoint of \(BC\).

1. Find the position vector of the point \(D\) such that \(\overrightarrow{BC} = \overrightarrow{AD}\). [3]
2. Find the magnitude of \(\overrightarrow{AM}\). [3]

The points \(A\) and \(B\) have position vectors \(\mathbf{i} - \mathbf{j} + \mathbf{k}\) and \(2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\) respectively, relative to the origin \(O\). The point \(C\) is on the line \(OA\) extended so that \(\overrightarrow{AC} = 2\overrightarrow{OA}\) and the point \(D\) is on the line \(OB\) extended so that \(\overrightarrow{BD} = 3\overrightarrow{OB}\). The point \(X\) is such that \(OCXD\) is a parallelogram.

1. Show that a vector equation of the line \(AX\) is \(\mathbf{r} = \mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(5\mathbf{i} + 7\mathbf{k})\) and find an equation of the line \(CD\) in a similar form. [5]
2. Prove that the lines \(AX\) and \(CD\) intersect and find the position vector of their point of intersection. [4]