1.10f Distance between points: using position vectors

10 \includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-4_561_599_744_774} The diagram shows triangle \(O A B\), in which the position vectors of \(A\) and \(B\) with respect to \(O\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = - 3 \mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k } .$$ \(C\) is a point on \(O A\) such that \(\overrightarrow { O C } = p \overrightarrow { O A }\), where \(p\) is a constant.

1. Find angle \(A O B\).
2. Find \(\overrightarrow { B C }\) in terms of \(p\) and vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Find the value of \(p\) given that \(B C\) is perpendicular to \(O A\).

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

1. In the case where \(O A B\) is a straight line, state the value of \(p\) and find the unit vector in the direction of \(\overrightarrow { O A }\).
2. In the case where \(O A\) is perpendicular to \(A B\), find the possible values of \(p\).
3. In the case where \(p = 3\), the point \(C\) is such that \(O A B C\) is a parallelogram. Find the position vector of \(C\).

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

1. In the case where \(p = 6\), find the unit vector in the direction of \(\overrightarrow { A B }\).
2. Find the values of \(p\) for which angle \(A O B = \cos ^ { - 1 } \left( \frac { 1 } { 5 } \right)\).

3 The point \(A\) has coordinates \(( 3,1 )\) and the point \(B\) has coordinates \(( - 21,11 )\). The point \(C\) is the mid-point of \(A B\).

1. Find the equation of the line through \(A\) that is perpendicular to \(y = 2 x - 7\).
2. Find the distance \(A C\).

6 Points \(A , B\) and \(C\) have coordinates \(A ( - 3,7 ) , B ( 5,1 )\) and \(C ( - 1 , k )\), where \(k\) is a constant.

1. Given that \(A B = B C\), calculate the possible values of \(k\). The perpendicular bisector of \(A B\) intersects the \(x\)-axis at \(D\).
2. Calculate the coordinates of \(D\).

7 Relative to an origin \(O\), the position vectors of points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 0 \\ 2 \\ - 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ 5 \\ - 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 3 \\ p \\ q \end{array} \right)$$

1. In the case where \(A B C\) is a straight line, find the values of \(p\) and \(q\).
2. In the case where angle \(B A C\) is \(90 ^ { \circ }\), express \(q\) in terms of \(p\).
3. In the case where \(p = 3\) and the lengths of \(A B\) and \(A C\) are equal, find the possible values of \(q\).

5 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { c } p - 6 \\ 2 p - 6 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { c } 4 - 2 p \\ p \\ 2 \end{array} \right)$$ where \(p\) is a constant.

1. For the case where \(O A\) is perpendicular to \(O B\), find the value of \(p\).
2. For the case where \(O A B\) is a straight line, find the vectors \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\). Find also the length of the line \(O A\).

5 The line \(\frac { x } { a } + \frac { y } { b } = 1\), where \(a\) and \(b\) are positive constants, intersects the \(x\) - and \(y\)-axes at the points \(A\) and \(B\) respectively. The mid-point of \(A B\) lies on the line \(2 x + y = 10\) and the distance \(A B = 10\). Find the values of \(a\) and \(b\).

9 \includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-16_533_601_258_772} The diagram shows a trapezium \(O A B C\) in which \(O A\) is parallel to \(C B\). The position vectors of \(A\) and \(B\) relative to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 2 \\ - 1 \end{array} \right)\) and \(\overrightarrow { O B } = \left( \begin{array} { l } 6 \\ 1 \\ 1 \end{array} \right)\).

1. Show that angle \(O A B\) is \(90 ^ { \circ }\).
   The magnitude of \(\overrightarrow { C B }\) is three times the magnitude of \(\overrightarrow { O A }\).
2. Find the position vector of \(C\).
3. Find the exact area of the trapezium \(O A B C\), giving your answer in the form \(a \sqrt { } b\), where \(a\) and \(b\) are integers.

9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 8 \\ - 6 \\ 5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 10 \\ 3 \\ - 13 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2 \\ - 3 \\ - 1 \end{array} \right)$$ A fourth point, \(D\), is such that the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\) are the first, second and third terms respectively of a geometric progression.

1. Find the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\).
2. Given that \(D\) is a point lying on the line through \(B\) and \(C\), find the two possible position vectors of the point \(D\).

3 Two points \(A\) and \(B\) have coordinates ( \(3 a , - a\) ) and ( \(- a , 2 a\) ) respectively, where \(a\) is a positive constant.

1. Find the equation of the line through the origin parallel to \(A B\).
2. The length of the line \(A B\) is \(3 \frac { 1 } { 3 }\) units. Find the value of \(a\).

8 \includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-12_595_748_260_699} The diagram shows a solid figure \(O A B C D E F\) having a horizontal rectangular base \(O A B C\) with \(O A = 6\) units and \(A B = 3\) units. The vertical edges \(O F , A D\) and \(B E\) have lengths 6 units, 4 units and 4 units respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O F\) respectively.

1. Find \(\overrightarrow { D F }\).
2. Find the unit vector in the direction of \(\overrightarrow { E F }\).
3. Use a scalar product to find angle \(E F D\).

6 \includegraphics[max width=\textwidth, alt={}, center]{d5d94eb8-7f41-4dff-b503-8be4f20e21b7-08_743_897_260_623} The diagram shows a solid figure \(O A B C D E F G\) with a horizontal rectangular base \(O A B C\) in which \(O A = 8\) units and \(A B = 6\) units. The rectangle \(D E F G\) lies in a horizontal plane and is such that \(D\) is 7 units vertically above \(O\) and \(D E\) is parallel to \(O A\). The sides \(D E\) and \(D G\) have lengths 4 units and 2 units respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively. Use a scalar product to find angle \(O B F\), giving your answer in the form \(\cos ^ { - 1 } \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.

7 \includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-12_784_677_260_735} The diagram shows a three-dimensional shape \(O A B C D E F G\). The base \(O A B C\) and the upper surface \(D E F G\) are identical horizontal rectangles. The parallelograms \(O A E D\) and \(C B F G\) both lie in vertical planes. Points \(P\) and \(Q\) are the mid-points of \(O D\) and \(G F\) respectively. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(\overrightarrow { O A }\) and \(\overrightarrow { O C }\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards. The position vectors of \(A , C\) and \(D\) are given by \(\overrightarrow { O A } = 6 \mathbf { i } , \overrightarrow { O C } = 8 \mathbf { j }\) and \(\overrightarrow { O D } = 2 \mathbf { i } + 10 \mathbf { k }\).

1. Express each of the vectors \(\overrightarrow { P B }\) and \(\overrightarrow { P Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Determine whether \(P\) is nearer to \(Q\) or to \(B\).
3. Use a scalar product to find angle \(B P Q\).

10 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } .$$ The line \(l\) has vector equation $$\mathbf { r } = ( 1 - 2 t ) \mathbf { i } + ( 5 + t ) \mathbf { j } + ( 2 - t ) \mathbf { k }$$

1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
2. The point \(P\) lies on \(l\) and is such that angle \(P A B\) is equal to \(60 ^ { \circ }\). Given that the position vector of \(P\) is \(( 1 - 2 t ) \mathbf { i } + ( 5 + t ) \mathbf { j } + ( 2 - t ) \mathbf { k }\), show that \(3 t ^ { 2 } + 7 t + 2 = 0\). Hence find the only possible position vector of \(P\).

9 With respect to the origin \(O\), the points \(A , B , C , D\) have position vectors given by $$\overrightarrow { O A } = \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \quad \overrightarrow { O B } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } , \quad \overrightarrow { O C } = 2 \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { O D } = - 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$$

1. Find the equation of the plane containing \(A , B\) and \(C\), giving your answer in the form \(a x + b y + c z = d\).
2. The line through \(D\) parallel to \(O A\) meets the plane with equation \(x + 2 y - z = 7\) at the point \(P\). Find the position vector of \(P\) and show that the length of \(D P\) is \(2 \sqrt { } ( 14 )\).

10 With respect to the origin \(O\), the points \(A , B , C , D\) have position vectors given by $$\overrightarrow { O A } = 4 \mathbf { i } + \mathbf { k } , \quad \overrightarrow { O B } = 5 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } , \quad \overrightarrow { O C } = \mathbf { i } + \mathbf { j } , \quad \overrightarrow { O D } = - \mathbf { i } - 4 \mathbf { k }$$

1. Calculate the acute angle between the lines \(A B\) and \(C D\).
2. Prove that the lines \(A B\) and \(C D\) intersect.
3. The point \(P\) has position vector \(\mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\). Show that the perpendicular distance from \(P\) to the line \(A B\) is equal to \(\sqrt { } 3\).

9
The diagram shows three points \(A , B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 1 \\ 2 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 0 \\ 3 \\ 1 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 3 \\ 0 \\ 4 \end{array} \right)\). The point \(D\) lies on \(B C\), between \(B\) and \(C\), and is such that \(C D = 2 D B\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the position vector of \(D\).
3. Show that the length of the perpendicular from \(A\) to \(O D\) is \(\frac { 1 } { 3 } \sqrt { } ( 65 )\).

7 The equations of two straight lines are $$\mathbf { r } = \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = a \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } + 3 a \mathbf { k } )$$ where \(a\) is a constant.

1. Show that the lines intersect for all values of \(a\).
2. Given that the point of intersection is at a distance of 9 units from the origin, find the possible values of \(a\).

7 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 4 \end{array} \right)$$ The plane \(m\) is perpendicular to \(A B\) and contains the point \(C\).

1. Find a vector equation for the line passing through \(A\) and \(B\).
2. Obtain the equation of the plane \(m\), giving your answer in the form \(a x + b y + c z = d\).
3. The line through \(A\) and \(B\) intersects the plane \(m\) at the point \(N\). Find the position vector of \(N\) and show that \(C N = \sqrt { } ( 13 )\).

10 The line \(l\) has vector equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\).

1. Find the position vectors of the two points on the line whose distance from the origin is \(\sqrt { } ( 10 )\).
2. The plane \(p\) has equation \(a x + y + z = 5\), where \(a\) is a constant. The acute angle between the line \(l\) and the plane \(p\) is equal to \(\sin ^ { - 1 } \left( \frac { 2 } { 3 } \right)\). Find the possible values of \(a\).

7 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 4 \end{array} \right)$$ The plane \(m\) is perpendicular to \(A B\) and contains the point \(C\).

1. Find a vector equation for the line passing through \(A\) and \(B\).
2. Obtain the equation of the plane \(m\), giving your answer in the form \(a x + b y + c z = d\).
3. The line through \(A\) and \(B\) intersects the plane \(m\) at the point \(N\). Find the position vector of \(N\) and show that \(C N = \sqrt { } ( 13 )\).

10 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 6 \mathbf { i } + 2 \mathbf { j }\) and \(\overrightarrow { O B } = 2 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\). The midpoint of \(O A\) is \(M\). The point \(N\) lying on \(A B\), between \(A\) and \(B\), is such that \(A N = 2 N B\).

1. Find a vector equation for the line through \(M\) and \(N\).
   The line through \(M\) and \(N\) intersects the line through \(O\) and \(B\) at the point \(P\).
2. Find the position vector of \(P\).
3. Calculate angle \(O P M\), giving your answer in degrees.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 2 \mathbf { i } - \mathbf { j }\) and \(\overrightarrow { O B } = \mathbf { j } - 2 \mathbf { k }\).

1. Show that \(O A = O B\) and use a scalar product to calculate angle \(A O B\) in degrees.
   The midpoint of \(A B\) is \(M\). The point \(P\) on the line through \(O\) and \(M\) is such that \(P A : O A = \sqrt { 7 } : 1\).
2. Find the possible position vectors of \(P\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 The lines \(l\) and \(m\) have equations $$\begin{aligned} l : & \mathbf { r } = a \mathbf { i } + 3 \mathbf { j } + b \mathbf { k } + \lambda ( c \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } ) \\ m : & \mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) \end{aligned}$$ Relative to the origin \(O\), the position vector of the point \(P\) is \(4 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\).

1. Given that \(l\) is perpendicular to \(m\) and that \(P\) lies on \(l\), find the values of the constants \(a , b\) and \(c\). [4]
2. The perpendicular from \(P\) meets line \(m\) at \(Q\). The point \(R\) lies on \(P Q\) extended, with \(P Q : Q R = 2 : 3\). Find the position vector of \(R\).