1.10f Distance between points: using position vectors

2 A particle, Q , moves so that its velocity, \(\mathbf { v }\), at time \(t\) is given by \(\mathbf { v } = ( 6 t - 6 ) \mathbf { i } + \left( 3 - 2 t + t ^ { 2 } \right) \mathbf { j } + 4 \mathbf { k }\), where \(0 \leqslant t \leqslant 6\).

1. Explain how you know that Q is never stationary. When Q is at a point A the direction of the acceleration of Q is parallel to the \(\mathbf { i }\) direction. When Q is at a point B the direction of the acceleration of Q makes an angle of \(45 ^ { \circ }\) with the \(\mathbf { i }\) direction.
2. Determine the straight-line distance AB .

3. Three forces \(( 4 \mathbf { i } - 7 \mathbf { j } + 9 \mathbf { k } ) \mathrm { N } , ( 5 \mathbf { i } + 3 \mathbf { j } - 8 \mathbf { k } ) \mathrm { N }\) and \(( - 2 \mathbf { i } + 6 \mathbf { j } - 11 \mathbf { k } ) \mathrm { N }\) act on a particle.

1. Find the resultant \(\mathbf { R }\) of the three forces.
2. The points \(A\) and \(B\) have position vectors \(( 3 \mathbf { i } + 4 \mathbf { j } - 12 \mathbf { k } ) \mathrm { m }\) and \(( a \mathbf { i } + 7 \mathbf { j } - 10 \mathbf { k } ) \mathrm { m }\) respectively, where \(a\) is a constant. The work done by \(\mathbf { R }\) in moving the particle from \(A\) to \(B\) is 21 J . Calculate the value of \(a\).
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1. \(\left[ \begin{array} { l } \text { With respect to the right-hand rule, a rotation through } \theta ^ { \circ } \text { anticlockwise about the } \\ y \text {-axis is represented by the matrix } \end{array} \right]\) \(\left( \begin{array} { c c c } \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ - \sin \theta & 0 & \cos \theta \end{array} \right)\)

The point \(P\) has coordinates (8, 3, 2)
The point \(Q\) is the image of \(P\) under the transformation reflection in the plane \(y = 0\)

1. Write down the coordinates of \(Q\) The point \(R\) is the image of \(P\) under the transformation rotation through \(120 ^ { \circ }\) anticlockwise about the \(y\)-axis, with respect to the right-hand rule.
2. Determine the exact coordinates of \(R\)
3. Hence find \(| \overrightarrow { P R } |\) giving your answer as a simplified surd.
4. Show that \(\overrightarrow { P R }\) and \(\overrightarrow { P Q }\) are perpendicular.
5. Hence determine the exact area of triangle \(P Q R\), giving your answer as a surd in simplest form.

1. Two birds are flying towards their nest, which is in a tree.

Relative to a fixed origin, the flight path of each bird is modelled by a straight line.
In the model, the equation for the flight path of the first bird is $$\mathbf { r } _ { 1 } = \left( \begin{array} { r } - 1 \\ 5 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ a \\ 0 \end{array} \right)$$ and the equation for the flight path of the second bird is $$\mathbf { r } _ { 2 } = \left( \begin{array} { r } 4 \\ - 1 \\ 3 \end{array} \right) + \mu \left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(a\) is a constant.
In the model, the angle between the birds' flight paths is \(120 ^ { \circ }\)

1. Determine the value of \(a\).
2. Verify that, according to the model, there is a common point on the flight paths of the two birds and find the coordinates of this common point. The position of the nest is modelled as being at this common point.
   The tree containing the nest is in a park.
   The ground level of the park is modelled by the plane with equation $$2 x - 3 y + z = 2$$
3. Hence determine the shortest distance from the nest to the ground level of the park.
4. By considering the model, comment on whether your answer to part (c) is reliable, giving a reason for your answer.

5 Points \(A , B\) and \(C\) have position vectors \(\left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) , \left( \begin{array} { c } 2 \\ - 1 \\ 5 \end{array} \right)\) and \(\left( \begin{array} { c } - 4 \\ 0 \\ 3 \end{array} \right)\) respectively.

1. Find the exact distance between the midpoint of \(A B\) and the midpoint of \(B C\). Point \(D\) has position vector \(\left( \begin{array} { c } x \\ - 6 \\ z \end{array} \right)\) and the line \(C D\) is parallel to the line \(A B\).
2. Find all the possible pairs of \(x\) and \(z\).

7. Two ships \(P\) and \(Q\) are travelling at night with constant velocities. At midnight, \(P\) is at the point with position vector \(( 20 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) relative to a fixed origin \(O\). At the same time, \(Q\) is at the point with position vector \(( 14 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km }\). Three hours later, \(P\) is at the point with position vector \(( 29 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\). The ship \(Q\) travels with velocity \(12 \mathbf { j } \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At time \(t\) hours after midnight, the position vectors of \(P\) and \(Q\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively. Find

1. the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\),
2. expressions for \(\mathbf { p }\) and \(\mathbf { q }\), in terms of \(t\), i and \(\mathbf { j }\). At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d \mathrm {~km}\).
3. By finding an expression for \(\overrightarrow { P Q }\), show that $$d ^ { 2 } = 25 t ^ { 2 } - 92 t + 292$$ Weather conditions are such that an observer on \(P\) can only see the lights on \(Q\) when the distance between \(P\) and \(Q\) is 15 km or less. Given that when \(t = 1\), the lights on \(Q\) move into sight of the observer,
4. find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer.
   1. In taking off, an aircraft moves on a straight runway \(A B\) of length 1.2 km . The aircraft moves from \(A\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It moves with constant acceleration and 20 s later it leaves the runway at \(C\) with speed \(74 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
   2. the acceleration of the aircraft,
   3. the distance \(B C\).
   4. Two small steel balls \(A\) and \(B\) have mass 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(B\) is twice the speed of \(A\). Find
   5. the speed of \(A\) immediately after the collision,
   6. the magnitude of the impulse exerted on \(B\) in the collision.
      
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6

1. 1. Express \(x ^ { 2 } - 8 x + 17\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 17\).
   3. State the value of \(x\) for which the minimum value of \(x ^ { 2 } - 8 x + 17\) occurs.
      (1 mark)
2. The point \(A\) has coordinates (5,4) and the point \(B\) has coordinates ( \(x , 7 - x\) ).
   1. Expand \(( x - 5 ) ^ { 2 }\).
   2. Show that \(A B ^ { 2 } = 2 \left( x ^ { 2 } - 8 x + 17 \right)\).
   3. Use your results from part (a) to find the minimum value of the distance \(A B\) as \(x\) varies.

7 The quadrilateral \(A B C D\) has vertices \(A ( 2,1,3 ) , B ( 6,5,3 ) , C ( 6,1 , - 1 )\) and \(D ( 2 , - 3 , - 1 )\).
The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left[ \begin{array} { r } 6 \\ 1 \\ - 1 \end{array} \right] + \lambda \left[ \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right]\).

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Show that the line \(A B\) is parallel to \(l _ { 1 }\).
   3. Verify that \(D\) lies on \(l _ { 1 }\).
2. The line \(l _ { 2 }\) passes through \(D ( 2 , - 3 , - 1 )\) and \(M ( 4,1,1 )\).
   1. Find the vector equation of \(l _ { 2 }\).
   2. Find the angle between \(l _ { 2 }\) and \(A C\).

6 The points \(A , B\) and \(C\) have coordinates \(( 3 , - 2,4 ) , ( 5,4,0 )\) and \(( 11,6 , - 4 )\) respectively.

1. 1. Find the vector \(\overrightarrow { B A }\).
   2. Show that the size of angle \(A B C\) is \(\cos ^ { - 1 } \left( - \frac { 5 } { 7 } \right)\).
2. The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 8 \\ - 3 \\ 2 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ 3 \\ - 2 \end{array} \right]\).
   1. Verify that \(C\) lies on \(l\).
   2. Show that \(A B\) is parallel to \(l\).
3. The quadrilateral \(A B C D\) is a parallelogram. Find the coordinates of \(D\).

6 The points \(A\) and \(B\) have coordinates \(( 2,4,1 )\) and \(( 3,2 , - 1 )\) respectively. The point \(C\) is such that \(\overrightarrow { O C } = 2 \overrightarrow { O B }\), where \(O\) is the origin.

1. Find the vectors:
   1. \(\overrightarrow { O C }\);
   2. \(\overrightarrow { A B }\).
2. 1. Show that the distance between the points \(A\) and \(C\) is 5 .
   2. Find the size of angle \(B A C\), giving your answer to the nearest degree.
3. The point \(P ( \alpha , \beta , \gamma )\) is such that \(B P\) is perpendicular to \(A C\). Show that \(4 \alpha - 3 \gamma = 15\).

7 The points \(A\) and \(B\) have coordinates ( \(3 , - 2,5\) ) and ( \(4,0,1\) ) respectively. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 6 \\ - 1 \\ 5 \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ - 1 \\ 4 \end{array} \right]\).

1. Find the distance between the points \(A\) and \(B\).
2. Verify that \(B\) lies on \(l _ { 1 }\).
   (2 marks)
3. The line \(l _ { 2 }\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { r } - 1 \\ 3 \\ - 8 \end{array} \right]\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(C\). Show that the points \(A , B\) and \(C\) form an isosceles triangle.
   (6 marks)

3.The lines \(L _ { 1 }\) and \(L _ { 2 }\) have the equations $$L _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ 9 \end{array} \right) + s \left( \begin{array} { l } 2 \\ p \\ 6 \end{array} \right) \quad \text { and } \quad L _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 15 \\ 12 \\ - 9 \end{array} \right) + t \left( \begin{array} { r } 4 \\ - 5 \\ 2 \end{array} \right)$$ where \(p\) is a constant.
The acute angle between \(L _ { 1 }\) and \(L _ { 2 }\) is \(\theta\) where \(\cos \theta = \frac { \sqrt { 5 } } { 3 }\)

1. Find the value of \(p\) . The line \(L _ { 3 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } - 15 \\ 12 \\ - 9 \end{array} \right) + u \left( \begin{array} { r } 8 \\ - 6 \\ - 5 \end{array} \right)\) and the lines \(L _ { 3 }\) and \(L _ { 2 }\) intersect at the point \(A\) .
   The point \(B\) on \(L _ { 2 }\) has position vector \(\left( \begin{array} { r } 5 \\ - 13 \\ 1 \end{array} \right)\) and point \(C\) lies on \(L _ { 3 }\) such that \(A B D C\) is a rhombus.
2. Find the two possible position vectors of \(D\) .

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

1. Show that the magnitude of \(\overrightarrow { O M }\) is equal to \(\sqrt { 17 }\). Point D is such that \(\overrightarrow { B C } = \overrightarrow { A D }\).
2. Show that position vector of the point D is \(11 \mathbf { i } - 8 \mathbf { j } - 6 \mathbf { k }\).

2 The points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j }\) and \(4 \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k }\) respectively.

1. Find the length of \(A B\). Point \(P\) has position vector \(p \mathbf { i } - 3 \mathbf { k }\), where \(p\) is a constant. \(P\) lies on the circumference of a circle of which \(A B\) is a diameter.
2. Find the two possible values of \(p\).

19

1. (ii) Verify that \(k = 0.8\) [0pt] [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) 19
2. Find the position vector of Amba when \(t = 4\) [0pt] [3 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) 19
3. At both \(t = 0\) and \(t = 4\) there is a distance of 5 metres between Jo and Amba's positions. Determine the shortest distance between their two parallel lines of motion.
   Fully justify your answer. \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-32_2492_1721_217_150}

1. Relative to a fixed origin \(O\)

• the point \(A\) has coordinates \(( - 10,5 , - 4 )\)
• the point \(B\) has coordinates \(( - 6,4 , - 1 )\)

The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).

1. Find a vector equation for \(l _ { 1 }\) The line \(l _ { 2 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3 \\ p \\ q \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 4 \\ 1 \end{array} \right)$$ where \(p\) and \(q\) are constants and \(\mu\) is a scalar parameter.
   Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at \(B\),
2. find the value of \(p\) and the value of \(q\). The acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\)
3. Find the exact value of \(\cos \theta\) Given that the point \(C\) lies on \(l _ { 2 }\) such that \(A C\) is perpendicular to \(l _ { 2 }\)
4. find the exact length of \(A C\), giving your answer as a surd.

9 Three points \(A , B\) and \(C\) have coordinates \(( 1,0,7 ) , ( 13,9,1 )\) and \(( 2 , - 1 , - 7 )\) respectively.

1. Use a scalar product to find angle \(A C B\).
2. Hence find the area of triangle \(A C B\).
3. Show that a vector equation of the line \(A B\) is given by \(\mathbf { r } = \mathbf { i } + 7 \mathbf { k } + \lambda ( 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )\), where \(\lambda\) is a scalar parameter.

10 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 5 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\).

1. Find the length of \(A B\).
2. Use a scalar product to find angle \(O A B\).

9 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 5 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\).

1. Find the length of \(A B\).
2. Use a scalar product to find angle \(O A B\).

9 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 5 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\).

1. Find the length of \(A B\).
2. Use a scalar product to find angle \(O A B\).

Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow{OA} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix} \text{ and } \overrightarrow{OB} = \begin{pmatrix} 4 \\ 1 \\ p \end{pmatrix}$$

1. Find the value of \(p\) for which \(\overrightarrow{OA}\) is perpendicular to \(\overrightarrow{OB}\). [2]
2. Find the values of \(p\) for which the magnitude of \(\overrightarrow{AB}\) is 7. [4]