1.10f Distance between points: using position vectors

7 A circle with centre \(( 5,2 )\) passes through the point \(( 7,5 )\).

1. Find an equation of the circle.
   The line \(y = 5 x - 10\) intersects the circle at \(A\) and \(B\).
2. Find the exact length of the chord \(A B\).

5 \includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-3_1070_754_255_699} The diagram shows a solid cylinder standing on a horizontal circular base, centre \(O\) and radius 4 units. The line \(B A\) is a diameter and the radius \(O C\) is at \(90 ^ { \circ }\) to \(O A\). Points \(O ^ { \prime } , A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\) lie on the upper surface of the cylinder such that \(O O ^ { \prime } , A A ^ { \prime } , B B ^ { \prime }\) and \(C C ^ { \prime }\) are all vertical and of length 12 units. The mid-point of \(B B ^ { \prime }\) is \(M\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O O ^ { \prime }\) respectively.

1. Express each of the vectors \(\overrightarrow { M O }\) and \(\overrightarrow { M C ^ { \prime } }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Hence find the angle \(O M C ^ { \prime }\).

7 The line \(L _ { 1 }\) has equation \(2 x + y = 8\). The line \(L _ { 2 }\) passes through the point \(A ( 7,4 )\) and is perpendicular to \(L _ { 1 }\).

1. Find the equation of \(L _ { 2 }\).
2. Given that the lines \(L _ { 1 }\) and \(L _ { 2 }\) intersect at the point \(B\), find the length of \(A B\).

9 Relative to an origin \(O\), the position vectors of the points \(A , B , C\) and \(D\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 1 \\ 3 \\ - 1 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ - 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { l } 4 \\ 2 \\ p \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } - 1 \\ 0 \\ q \end{array} \right) ,$$ where \(p\) and \(q\) are constants. Find

1. the unit vector in the direction of \(\overrightarrow { A B }\),
2. the value of \(p\) for which angle \(A O C = 90 ^ { \circ }\),
3. the values of \(q\) for which the length of \(\overrightarrow { A D }\) is 7 units.

11 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$$

1. Use a scalar product to find angle \(A O B\), correct to the nearest degree.
2. Find the unit vector in the direction of \(\overrightarrow { A B }\).
3. The point \(C\) is such that \(\overrightarrow { O C } = 6 \mathbf { j } + p \mathbf { k }\), where \(p\) is a constant. Given that the lengths of \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\) are equal, find the possible values of \(p\).

5 The curve \(y ^ { 2 } = 12 x\) intersects the line \(3 y = 4 x + 6\) at two points. Find the distance between the two points.

8 \includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-3_517_1117_1362_514} The diagram shows the roof of a house. The base of the roof, \(O A B C\), is rectangular and horizontal with \(O A = C B = 14 \mathrm {~m}\) and \(O C = A B = 8 \mathrm {~m}\). The top of the roof \(D E\) is 5 m above the base and \(D E = 6 \mathrm {~m}\). The sloping edges \(O D , C D , A E\) and \(B E\) are all equal in length. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O C\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards.

1. Express the vector \(\overrightarrow { O D }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\), and find its magnitude.
2. Use a scalar product to find angle \(D O B\).

10 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and \(3 \mathbf { i } - 2 \mathbf { j } + p \mathbf { k }\) respectively.

1. Find the value of \(p\) for which \(O A\) and \(O B\) are perpendicular.
2. In the case where \(p = 6\), use a scalar product to find angle \(A O B\), correct to the nearest degree.
3. Express the vector \(\overrightarrow { A B }\) is terms of \(p\) and hence find the values of \(p\) for which the length of \(A B\) is 3.5 units.

11 \includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-4_563_965_813_591} In the diagram, the points \(A\) and \(C\) lie on the \(x\) - and \(y\)-axes respectively and the equation of \(A C\) is \(2 y + x = 16\). The point \(B\) has coordinates ( 2,2 ). The perpendicular from \(B\) to \(A C\) meets \(A C\) at the point \(X\).

1. Find the coordinates of \(X\). The point \(D\) is such that the quadrilateral \(A B C D\) has \(A C\) as a line of symmetry.
2. Find the coordinates of \(D\).
3. Find, correct to 1 decimal place, the perimeter of \(A B C D\).

10 \includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-4_552_629_842_758} The diagram shows the parallelogram \(O A B C\). Given that \(\overrightarrow { O A } = \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { O C } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k }\), find

1. the unit vector in the direction of \(\overrightarrow { O B }\),
2. the acute angle between the diagonals of the parallelogram,
3. the perimeter of the parallelogram, correct to 1 decimal place.

6 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } + 8 \mathbf { k } , \quad \overrightarrow { O C } = - \mathbf { i } - 2 \mathbf { j } + 10 \mathbf { k }$$

1. Use a scalar product to find angle \(A B C\).
2. Find the perimeter of triangle \(A B C\), giving your answer correct to 2 decimal places.

9 The coordinates of \(A\) are \(( - 3,2 )\) and the coordinates of \(C\) are (5,6). The mid-point of \(A C\) is \(M\) and the perpendicular bisector of \(A C\) cuts the \(x\)-axis at \(B\).

1. Find the equation of \(M B\) and the coordinates of \(B\).
2. Show that \(A B\) is perpendicular to \(B C\).
3. Given that \(A B C D\) is a square, find the coordinates of \(D\) and the length of \(A D\).

6 Relative to an origin \(O\), the position vectors of three points, \(A , B\) and \(C\), are given by $$\overrightarrow { O A } = \mathbf { i } + 2 p \mathbf { j } + q \mathbf { k } , \quad \overrightarrow { O B } = q \mathbf { j } - 2 p \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = - \left( 4 p ^ { 2 } + q ^ { 2 } \right) \mathbf { i } + 2 p \mathbf { j } + q \mathbf { k }$$ where \(p\) and \(q\) are constants.

1. Show that \(\overrightarrow { O A }\) is perpendicular to \(\overrightarrow { O C }\) for all non-zero values of \(p\) and \(q\).
2. Find the magnitude of \(\overrightarrow { C A }\) in terms of \(p\) and \(q\).
3. For the case where \(p = 3\) and \(q = 2\), find the unit vector parallel to \(\overrightarrow { B A }\).

6 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 3 \mathbf { i } + p \mathbf { j } + q \mathbf { k }$$ where \(p\) and \(q\) are constants.

1. State the values of \(p\) and \(q\) for which \(\overrightarrow { O A }\) is parallel to \(\overrightarrow { O B }\).
2. In the case where \(q = 2 p\), find the value of \(p\) for which angle \(B O A\) is \(90 ^ { \circ }\).
3. In the case where \(p = 1\) and \(q = 8\), find the unit vector in the direction of \(\overrightarrow { A B }\).

7 The point \(A\) has coordinates \(( p , 1 )\) and the point \(B\) has coordinates \(( 9,3 p + 1 )\), where \(p\) is a constant.

1. For the case where the distance \(A B\) is 13 units, find the possible values of \(p\).
2. For the case in which the line with equation \(2 x + 3 y = 9\) is perpendicular to \(A B\), find the value of \(p\).

10 Relative to an origin \(O\), the position vectors of points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ 1 \\ - 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 5 \\ - 1 \\ k \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2 \\ 6 \\ - 3 \end{array} \right)$$ respectively, where \(k\) is a constant.

1. Find the value of \(k\) in the case where angle \(A O B = 90 ^ { \circ }\).
2. Find the possible values of \(k\) for which the lengths of \(A B\) and \(O C\) are equal. The point \(D\) is such that \(\overrightarrow { O D }\) is in the same direction as \(\overrightarrow { O A }\) and has magnitude 9 units. The point \(E\) is such that \(\overrightarrow { O E }\) is in the same direction as \(\overrightarrow { O C }\) and has magnitude 14 units.
3. Find the magnitude of \(\overrightarrow { D E }\) in the form \(\sqrt { } n\) where \(n\) is an integer.

9 The position vectors of \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { c } 1 \\ 5 \\ p \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right) ,$$ where \(p\) is a constant.

1. Find the value of \(p\) for which the lengths of \(A B\) and \(C B\) are equal.
2. For the case where \(p = 1\), use a scalar product to find angle \(A B C\).

11 Triangle \(A B C\) has vertices at \(A ( - 2 , - 1 ) , B ( 4,6 )\) and \(C ( 6 , - 3 )\).

1. Show that triangle \(A B C\) is isosceles and find the exact area of this triangle.
2. The point \(D\) is the point on \(A B\) such that \(C D\) is perpendicular to \(A B\). Calculate the \(x\)-coordinate of \(D\).

2 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 6 \\ p \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 6 \\ - 7 \end{array} \right)$$ and angle \(A O B = 90 ^ { \circ }\).

1. Find the value of \(p\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) The point \(C\) is such that \(\overrightarrow { O C } = \frac { 2 } { 3 } \overrightarrow { O A }\).
2. Find the unit vector in the direction of \(\overrightarrow { B C }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

5 Two points have coordinates \(A ( 5,7 )\) and \(B ( 9 , - 1 )\).

1. Find the equation of the perpendicular bisector of \(A B\). The line through \(C ( 1,2 )\) parallel to \(A B\) meets the perpendicular bisector of \(A B\) at the point \(X\).
2. Find, by calculation, the distance \(B X\).

7 \includegraphics[max width=\textwidth, alt={}, center]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-3_529_698_260_721} The diagram shows a pyramid \(O A B C\) with a horizontal triangular base \(O A B\) and vertical height \(O C\). Angles \(A O B , B O C\) and \(A O C\) are each right angles. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O C\) respectively, with \(O A = 4\) units, \(O B = 2.4\) units and \(O C = 3\) units. The point \(P\) on \(C A\) is such that \(C P = 3\) units.

1. Show that \(\overrightarrow { C P } = 2.4 \mathbf { i } - 1.8 \mathbf { k }\).
2. Express \(\overrightarrow { O P }\) and \(\overrightarrow { B P }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Use a scalar product to find angle \(B P C\).

4 Relative to an origin \(O\), the position vectors of points \(P\) and \(Q\) are given by $$\overrightarrow { O P } = \left( \begin{array} { r } - 2 \\ 3 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O Q } = \left( \begin{array} { l } 2 \\ 1 \\ q \end{array} \right)$$ where \(q\) is a constant.

1. In the case where \(q = 3\), use a scalar product to show that \(\cos P O Q = \frac { 1 } { 7 }\).
2. Find the values of \(q\) for which the length of \(\overrightarrow { P Q }\) is 6 units.
   \includegraphics[max width=\textwidth, alt={}]{933cdfe1-27bb-450d-8b9a-b494916242cb-3_647_741_845_699}
   The diagram shows the cross-section of a hollow cone and a circular cylinder. The cone has radius 6 cm and height 12 cm , and the cylinder has radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The cylinder just fits inside the cone with all of its upper edge touching the surface of the cone.

10 \includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-4_597_693_274_726} The diagram shows a cube \(O A B C D E F G\) in which the length of each side is 4 units. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively. The mid-points of \(O A\) and \(D G\) are \(P\) and \(Q\) respectively and \(R\) is the centre of the square face \(A B F E\).

1. Express each of the vectors \(\overrightarrow { P R }\) and \(\overrightarrow { P Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to find angle \(Q P R\).
3. Find the perimeter of triangle \(P Q R\), giving your answer correct to 1 decimal place.

4 \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-2_558_1488_863_331} The diagram shows a semicircular prism with a horizontal rectangular base \(A B C D\). The vertical ends \(A E D\) and \(B F C\) are semicircles of radius 6 cm . The length of the prism is 20 cm . The mid-point of \(A D\) is the origin \(O\), the mid-point of \(B C\) is \(M\) and the mid-point of \(D C\) is \(N\). The points \(E\) and \(F\) are the highest points of the semicircular ends of the prism. The point \(P\) lies on \(E F\) such that \(E P = 8 \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O D , O M\) and \(O E\) respectively.

1. Express each of the vectors \(\overrightarrow { P A }\) and \(\overrightarrow { P N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to calculate angle \(A P N\).

9 \includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-4_582_1072_255_541} The diagram shows a pyramid \(O A B C P\) in which the horizontal base \(O A B C\) is a square of side 10 cm and the vertex \(P\) is 10 cm vertically above \(O\). The points \(D , E , F , G\) lie on \(O P , A P , B P , C P\) respectively and \(D E F G\) is a horizontal square of side 6 cm . The height of \(D E F G\) above the base is \(a \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively.

1. Show that \(a = 4\).
2. Express the vector \(\overrightarrow { B G }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Use a scalar product to find angle \(G B A\).