1.10f Distance between points: using position vectors

The point \(A\) has coordinates \((4, -3, 2)\). The line \(l_1\) passes through \(A\) and has equation \(\mathbf{r} = \begin{bmatrix} 4 \\ -3 \\ 2 \end{bmatrix} + \lambda \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}\). The line \(l_2\) has equation \(\mathbf{r} = \begin{bmatrix} -1 \\ 3 \\ 4 \end{bmatrix} + \mu \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix}\). The point \(B\) lies on \(l_2\) where \(\mu = 2\).

1. Find the vector \(\overrightarrow{AB}\). [3 marks]
2. 1. Show that the lines \(l_1\) and \(l_2\) intersect. [4 marks]
   2. The lines \(l_1\) and \(l_2\) intersect at the point \(P\). Find the coordinates of \(P\). [1 mark]
3. The point \(C\) lies on a line which is parallel to \(l_1\) and which passes through the point \(B\). The points \(A\), \(B\), \(C\) and \(P\) are the vertices of a parallelogram. Find the coordinates of the two possible positions of the point \(C\). [4 marks]

Fig. 7 shows a tetrahedron ABCD. The coordinates of the vertices, with respect to axes Oxyz, are A(-3, 0, 0), B(2, 0, -2), C(0, 4, 0) and D(0, 4, 5). \includegraphics{figure_7}

1. Find the lengths of the edges AB and AC, and the size of the angle CAB. Hence calculate the area of triangle ABC. [7]
2. 1. Verify that 4i - 3j + 10k is normal to the plane ABC. [2]
   2. Hence find the equation of this plane. [2]
3. Write down a vector equation for the line through D perpendicular to the plane ABC. Hence find the point of intersection of this line with the plane ABC. [5]

The volume of a tetrahedron is \(\frac{1}{3} \times \text{area of base} \times \text{height}\).

1. Find the volume of the tetrahedron ABCD. [2]

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]

A particle \(P\) moves with a constant velocity \((3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) with respect to a fixed origin \(O\). It passes through the point \(A\) whose position vector is \((2\mathbf{i} + 11\mathbf{j})\) m at \(t = 0\).

1. Find the angle in degrees that the velocity vector of \(P\) makes with the vector \(\mathbf{i}\). [2 marks]
2. Calculate the distance of \(P\) from \(O\) when \(t = 2\). [4 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 tetrahedron \(ABCD\) is such that \(AB\) is perpendicular to the base \(BCD\). The coordinates of the points \(A\), \(C\) and \(D\) are \((-1, -7, 2)\), \((5, 0, 3)\) and \((-1, 3, 3)\) respectively, and the equation of the plane \(BCD\) is \(x + 2y - 2z = -1\).

1. Find, in either order, the coordinates of \(B\) and the length of \(AB\). [5]
2. Find the acute angle between the planes \(ACD\) and \(BCD\). [6]

The points \(O\) and \(A\) have position vectors \(\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 0 \\ 8 \end{pmatrix}\) respectively. The point \(P\) is such that \(\overrightarrow{OP} = k\overrightarrow{OA}\), where \(k\) is a non-zero constant.

1. Find, in terms of \(k\), the length of \(OP\). [1] Point \(B\) has position vector \(\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}\) and angle \(OPB\) is a right angle.
2. Determine the value of \(k\). [4]

Two particles, \(A\) and \(B\), lie at rest on a smooth horizontal plane. \(A\) has position vector \(\mathbf{r}_A = (13\mathbf{i} - 22\mathbf{j})\) metres \(B\) has position vector \(\mathbf{r}_B = (3\mathbf{i} + 2\mathbf{j})\) metres

1. Calculate the distance between \(A\) and \(B\). [2 marks]
2. Three forces, \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are applied to particle \(A\), where \(\mathbf{F}_1 = (-2\mathbf{i} + 4\mathbf{j})\) newtons \(\mathbf{F}_2 = (6\mathbf{i} - 10\mathbf{j})\) newtons Given that \(A\) remains at rest, explain why \(\mathbf{F}_3 = (-4\mathbf{i} + 6\mathbf{j})\) newtons [1 mark]
3. A force of \((5\mathbf{i} - 12\mathbf{j})\) newtons, is applied to \(B\), so that \(B\) moves, from rest, in a straight line towards \(A\). \(B\) has a mass of \(0.8 \text{kg}\)
   1. Show that the acceleration of \(B\) towards \(A\) is \(16.25 \text{m s}^{-2}\) [2 marks]
   2. Hence, find the time taken for \(B\) to reach \(A\). Give your answer to two significant figures. [2 marks]

One of the following is an expression for the distance between the points represented by position vectors \(5\mathbf{i} - 3\mathbf{j}\) and \(18\mathbf{i} + 7\mathbf{j}\) Identify the correct expression. Tick (\(\checkmark\)) one box. [1 mark] \(\sqrt{13^2 + 4^2}\) \(\sqrt{13^2 + 10^2}\) \(\sqrt{23^2 + 4^2}\) \(\sqrt{23^2 + 10^2}\)

It is given that two points \(A\) and \(B\) have position vectors $$\overrightarrow{OA} = \begin{bmatrix} 5 \\ -1 \end{bmatrix} \text{ metres} \quad \text{and} \quad \overrightarrow{OB} = \begin{bmatrix} 13 \\ 5 \end{bmatrix} \text{ metres.}$$

1. Show that the distance from \(A\) to \(B\) is 10 metres. [3 marks]
2. A constant resultant force, of magnitude \(R\) newtons, acts on a particle so that it moves in a straight line passing through the same two points \(A\) and \(B\) At \(A\), the speed of the particle is 3 m s\(^{-1}\) in the direction from \(A\) to \(B\) The particle takes 2 seconds to travel from \(A\) to \(B\) The mass of the particle is 150 grams. Find the value of \(R\) [3 marks]

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]

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]

A circle \(C\) with centre at \((-2, 6)\) passes through the point \((10, 11)\).

1. Show that the circle \(C\) also passes through the point \((10, 1)\). [3]

The tangent to the circle \(C\) at the point \((10, 11)\) meets the \(y\) axis at the point \(P\) and the tangent to the circle \(C\) at the point \((10, 1)\) meets the \(y\) axis at the point \(Q\).

1. Show that the distance \(PQ\) is \(58\) explaining your method clearly. [7]

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 \(t = 0\) s, the position vector of an object \(A\) is \(\mathbf{i}\) m and the position vector of another object \(B\) is \(3\mathbf{i}\) m. The constant velocity vector of \(A\) is \(2\mathbf{i} + 5\mathbf{j} - 4k\) ms\(^{-1}\) and the constant velocity vector of \(B\) is \(\mathbf{i} + 3\mathbf{j} - 5k\) ms\(^{-1}\). Determine the value of \(t\) when \(A\) and \(B\) are closest together and find the least distance between \(A\) and \(B\). [9]

\includegraphics{figure_2} Figure 2 is a sketch showing the line \(l_1\) with equation \(y = 2x - 1\) and the point \(A\) with coordinates \((-2, 3)\). The line \(l_2\) passes through \(A\) and is perpendicular to \(l_1\)

1. Find the equation of \(l_2\) writing your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants to be found. [3]

The point \(B\) and the point \(C\) lie on \(l_1\) such that \(ABC\) is an isosceles triangle with \(AB = AC = 2\sqrt{13}\)

1. Show that the \(x\) coordinates of points \(B\) and \(C\) satisfy the equation $$5x^2 - 12x - 32 = 0$$ [4]

Given that \(B\) lies in the 3rd quadrant

1. find, using algebra and showing your working, the coordinates of \(B\). [4]

The position vector of point \(A\) is \(\mathbf{a} = -9\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}\). The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf{a}\).

1. Determine the shortest distance between the origin, \(O\), and \(l\). [2] \(l\) is also perpendicular to the vector \(\mathbf{b}\) where \(\mathbf{b} = -2\mathbf{i} + \mathbf{j} + \mathbf{k}\).
2. Find a vector which is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\). [1]
3. Write down an equation of \(l\) in vector form. [1] \(P\) is a point on \(l\) such that \(PA = 2OA\).
4. Find angle \(POA\) giving your answer to 3 significant figures. [3] \(C\) is a point whose position vector, \(\mathbf{c}\), is given by \(\mathbf{c} = p\mathbf{a}\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf{r} = \mathbf{c} + \mu\mathbf{b}\). The point with position vector \(9\mathbf{i} + 8\mathbf{j} - 12\mathbf{k}\) lies on \(m\).
5. Find the value of \(p\). [3]

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

1. Find \(|\vec{AB}|\) giving your answer as a simplified surd. [2] Given that \(ABCD\) is a parallelogram,
2. find the position vector of the point \(D\). [2] The point \(E\) is positioned such that • \(ACE\) is a straight line • \(AC : CE = 2 : 1\)
3. Find the coordinates of the point \(E\). [2]

The equations of two intersecting lines \(l_1\) and \(l_2\) are $$l_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ a \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 7 \\ 9 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}$$ where \(a\) is a constant. The equation of the plane \(\Pi\) is $$\mathbf{r} \cdot \begin{pmatrix} 1 \\ 5 \\ 3 \end{pmatrix} = -14.$$ \(l_1\) and \(\Pi\) intersect at \(Q\). \(l_2\) and \(\Pi\) intersect at \(R\).

1. Verify that the coordinates of \(R\) are \((13, 3, -14)\). [2]
2. Determine the exact value of the length of \(QR\). [7]

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]