1.10f Distance between points: using position vectors

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of a parallelogram \(X A P B\).
Given that \(\overrightarrow { O X } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right)\) $$\begin{aligned} & \overrightarrow { O A } = \left( \begin{array} { l } 0 \\ 4 \\ 1 \end{array} \right) \\ & \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 3 \\ 1 \end{array} \right) \end{aligned}$$ a. Find the coordinates of the point \(P\).
b. Show that \(X A P B\) is a rhombus.
c. Find the exact area of the rhombus \(X A P B\).

2. Relative to a fixed origin \(O\),
the point \(A\) has position vector \(( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\) the point \(B\) has position vector ( \(\mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k }\) )
and the point \(C\) has position vector \(( - \mathbf { i } + \mathbf { j } + a \mathbf { k } )\), where \(a\) is a constant and \(a > 0\).
Given that \(| \overrightarrow { B C } | = \sqrt { 41 }\) a. show that \(a = 2\). \(D\) is the point such that \(A B C D\) forms a parallelogram.
b. Find the position vector of \(D\).

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

• the point \(P\) has position vector \(( 0 , - 1,2 )\)
• the point \(Q\) has position vector \(( 1,1,5 )\)
• the point \(R\) has position vector ( \(3,5 , m\) )
  where \(m\) is a constant.
  Given that \(P , Q\) and \(R\) lie on a straight line,
  a. find the value of \(m\)

The line segment \(O Q\) is extended to a point \(T\) so that \(\overrightarrow { R T }\) is parallel to \(\overrightarrow { O P }\) b. Show that \(| \overrightarrow { O T } | = 9 \sqrt { 3 }\).

Relative to a fixed origin \(O\),
the point \(A\) has position vector \(( 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } )\),
the point \(B\) has position vector ( \(4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }\) ),
and the point \(C\) has position vector ( \(a \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k }\) ), where \(a\) is a constant and \(a < 0 D\) is the point such that \(\overrightarrow { A B } = \overrightarrow { B D }\).

1. Find the position vector of \(D\). Given \(| \overrightarrow { A C } | = 4\)
2. find the value of \(a\).

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

• the point \(A\) has position vector \(4 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\)
• the point \(B\) has position vector \(4 \mathbf { j } + 6 \mathbf { k }\)
• the point \(C\) has position vector \(- 16 \mathbf { i } + p \mathbf { j } + 10 \mathbf { k }\) where \(p\) is a constant.
  Given that \(A , B\) and \(C\) lie on a straight line,
  1. find the value of \(p\).

The line segment \(O B\) is extended to a point \(D\) so that \(\overrightarrow { C D }\) is parallel to \(\overrightarrow { O A }\) (b) Find \(| \overrightarrow { O D } |\), writing your answer as a fully simplified surd.

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

• \(A\) is the point with position vector \(12 \mathbf { i }\)
• \(B\) is the point with position vector \(16 \mathbf { j }\)
• \(C\) is the point with position vector \(( 50 \mathbf { i } + 136 \mathbf { j } )\)
• \(D\) is the point with position vector \(( 22 \mathbf { i } + 24 \mathbf { j } )\)
  1. Show that \(A D\) is parallel to \(B C\).

Points \(A , B , C\) and \(D\) are used to model the vertices of a running track in the shape of a quadrilateral. Runners complete one lap by running along all four sides of the track.
The lengths of the sides are measured in metres. Given that a particular runner takes exactly 5 minutes to complete 2 laps,

calculate the average speed of this runner, giving the answer in kilometres per hour.

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the straight line \(l\).
Line \(l\) passes through the points \(A\) and \(B\).
Relative to a fixed origin \(O\)

• the point \(A\) has position vector \(2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 6 \mathbf { j } + 8 \mathbf { k }\)
  1. Find \(\overrightarrow { A B }\)

Given that a point \(P\) lies on \(l\) such that $$| \overrightarrow { A P } | = 2 | \overrightarrow { B P } |$$

find the possible position vectors of \(P\).

5 Points \(A , B , C\) and \(D\) have position vectors \(\mathbf { a } = \binom { 1 } { 2 } , \mathbf { b } = \binom { 3 } { 5 } , \mathbf { c } = \binom { 7 } { 4 }\) and \(\mathbf { d } = \binom { 4 } { k }\).

1. Find the value of \(k\) for which \(D\) is the midpoint of \(A C\).
2. Find the two values of \(k\) for which \(| \overrightarrow { A D } | = \sqrt { 13 }\).
3. Find one value of \(k\) for which the four points form a trapezium.

6 The vertices of triangle \(A B C\) are \(A ( - 3,1 ) , B ( 5,0 )\) and \(C ( 9,7 )\).

1. Show that \(A B = B C\).
2. Show that angle \(A B C\) is not a right angle.
3. Find the coordinates of the midpoint of \(A C\).
4. Determine the equation of the line of symmetry of the triangle, giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers to be determined.
5. Write down an equation of the circle with centre \(A\) which passes through \(B\). This circle intersects the line of symmetry of the triangle at \(B\) and at a second point.
6. Find the coordinates of this second point.

3 The points \(A\) and \(B\) have position vectors \(\binom { 2 } { - 1 }\) and \(\binom { 5 } { 4 }\) respectively. The vector \(\overrightarrow { \mathrm { AC } }\) is \(\binom { - 2 } { 2 }\).

1. Write down the position vector of C as a column vector.
2. Show that B is equidistant from A and C .

8 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 16 y + 48 = 0\).

1. Find the coordinates of C . A line has equation \(\mathrm { y } = \mathrm { x } - 2\) and intersects the circle at the points A and B . The midpoints of AC and BC are \(\mathrm { A } ^ { \prime }\) and \(\mathrm { B } ^ { \prime }\) respectively.
2. Determine the exact distance \(\mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime }\).

6 Two points, \(A\) and \(B\), have position vectors \(\mathbf { a } = \mathbf { i } - 3 \mathbf { j }\) and \(\mathbf { b } = 4 \mathbf { i } + 3 \mathbf { j }\).
The point C lies on the line \(y = 1\). The lengths of the line segments AC and BC are equal. Determine the position vector of \(C\).

4 The position vector of \(P\) is \(\mathbf { p } = \binom { 4 } { 3 }\) and the position vector of \(Q\) is \(\mathbf { q } = \binom { 28 } { 10 }\).

1. Determine the magnitude of \(\overrightarrow { \mathrm { PQ } }\).
2. Determine the angle between \(\overrightarrow { \mathrm { PQ } }\) and the positive \(x\)-direction.

5 Show that the distance between the points \(( 5,2 )\) and \(( 11 , - 1 )\) is \(a \sqrt { b }\), where \(a\) and \(b\) are integers to be determined.

8 With respect to an origin O , the position vectors of the points A and B are \(\overrightarrow { \mathrm { OA } } = \binom { - 3 } { 20 }\) and \(\overrightarrow { \mathrm { OB } } = \binom { 6 } { 8 }\).

1. Determine whether \(| \overrightarrow { \mathrm { AB } } | > 200\). The point C is such that \(\overrightarrow { \mathrm { AC } } = \binom { 18 } { - 24 }\).
2. Determine whether \(\mathrm { A } , \mathrm { B }\) and C are collinear.

3 The points \(A\) and \(B\) have position vectors \(\mathbf { a } = \left( \begin{array} { r } 3 \\ 2 \\ - 1 \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { r } - 1 \\ 4 \\ 8 \end{array} \right)\) respectively.
Show that the exact value of the distance \(A B\) is \(\sqrt { \mathbf { 1 0 1 } }\).

5 You are given that \(\overrightarrow { \mathrm { OA } } = \binom { 3 } { - 1 }\) and \(\overrightarrow { \mathrm { OB } } = \binom { 5 } { - 3 }\). Determine the exact length of \(A B\).

1 Calculate the exact distance between the points ( \(2 , - 1\) ) and ( 6,1 ). Give your answer in the form \(\mathrm { a } \sqrt { \mathrm { b } }\), where \(a\) and \(b\) are prime numbers.

10 Point A has position vector \(\left( \begin{array} { l } a \\ b \\ 0 \end{array} \right)\) where \(a\) and \(b\) can vary, point B has position vector \(\left( \begin{array} { l } 4 \\ 2 \\ 0 \end{array} \right)\) and point C has position vector \(\left( \begin{array} { l } 2 \\ 4 \\ 2 \end{array} \right)\). ABC is an isosceles triangle with \(\mathrm { AC } = \mathrm { AB }\).

1. Show that \(a - b + 1 = 0\).
2. Determine the position vector of A such that triangle ABC has minimum area. Answer all the questions.
   Section B (15 marks) The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

5 A circle with centre \(C ( - 5,6 )\) touches the \(y\)-axis, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-6_444_698_372_680}

1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. 1. Verify that the point \(P ( - 2,2 )\) lies on the circle.
   2. Find an equation of the normal to the circle at the point \(P\).
   3. The mid-point of \(P C\) is \(M\). Determine whether the point \(P\) is closer to the point \(M\) or to the origin \(O\).

6 The circle with centre \(C ( 5,8 )\) touches the \(y\)-axis, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-5_485_631_370_715}

1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. 1. Verify that the point \(A ( 2,12 )\) lies on the circle.
   2. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(s x + t y + u = 0\), where \(s , t\) and \(u\) are integers.
3. The points \(P\) and \(Q\) lie on the circle, and the mid-point of \(P Q\) is \(M ( 7,12 )\).
   1. Show that the length of \(C M\) is \(n \sqrt { 5 }\), where \(n\) is an integer.
   2. Hence find the area of triangle \(P C Q\).

5

1. 1. Express \(2 x ^ { 2 } + 6 x + 5\) in the form \(2 ( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
   2. Hence write down the minimum value of \(2 x ^ { 2 } + 6 x + 5\).
2. The point \(A\) has coordinates \(( - 3,5 )\) and the point \(B\) has coordinates \(( x , 3 x + 9 )\).
   1. Show that \(A B ^ { 2 } = 5 \left( 2 x ^ { 2 } + 6 x + 5 \right)\).
   2. Use your result from part (a)(ii) to find the minimum value of the length \(A B\) as \(x\) varies, giving your answer in the form \(\frac { 1 } { 2 } \sqrt { n }\), where \(n\) is an integer.

6. The straight line \(l\) has the equation \(x - 2 y = 12\) and meets the coordinate axes at the points \(A\) and \(B\). Find the distance of the mid-point of \(A B\) from the origin, giving your answer in the form \(k \sqrt { 5 }\).

6

1. The points \(A , B\) and \(C\) have coordinates \(( 3,1 , - 6 ) , ( 5 , - 2,0 )\) and \(( 8 , - 4 , - 6 )\) respectively.
   1. Show that the vector \(\overrightarrow { A C }\) is given by \(\overrightarrow { A C } = n \left[ \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right]\), where \(n\) is an integer.
   2. Show that the acute angle \(A C B\) is given by \(\cos ^ { - 1 } \left( \frac { 5 \sqrt { 2 } } { 14 } \right)\).
2. Find a vector equation of the line \(A C\).
3. The point \(D\) has coordinates \(( 6 , - 1 , p )\). It is given that the lines \(A C\) and \(B D\) intersect.
   1. Find the value of \(p\).
   2. Show that \(A B C D\) is a rhombus, and state the length of each of its sides.

5 The points \(A\) and \(B\) have coordinates \(( 5,1 , - 2 )\) and \(( 4 , - 1,3 )\) respectively.
The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } - 8 \\ 5 \\ - 6 \end{array} \right] + \mu \left[ \begin{array} { r } 5 \\ 0 \\ - 2 \end{array} \right]\).

1. Find a vector equation of the line that passes through \(A\) and \(B\).
2. 1. Show that the line that passes through \(A\) and \(B\) intersects the line \(l\), and find the coordinates of the point of intersection, \(P\).
   2. The point \(C\) lies on \(l\) such that triangle \(P B C\) has a right angle at \(B\). Find the coordinates of \(C\).