Perpendicularity conditions

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.

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

1. Find the value of \(p\) for which \(\overrightarrow { O A }\) is perpendicular to \(\overrightarrow { O B }\).
2. Find the values of \(p\) for which the magnitude of \(\overrightarrow { A B }\) is 7 .

6 Two vectors \(\mathbf { u }\) and \(\mathbf { v }\) are such that \(\mathbf { u } = \left( \begin{array} { c } p ^ { 2 } \\ - 2 \\ 6 \end{array} \right)\) and \(\mathbf { v } = \left( \begin{array} { c } 2 \\ p - 1 \\ 2 p + 1 \end{array} \right)\), where \(p\) is a constant.

1. Find the values of \(p\) for which \(\mathbf { u }\) is perpendicular to \(\mathbf { v }\).
2. For the case where \(p = 1\), find the angle between the directions of \(\mathbf { u }\) and \(\mathbf { v }\).

4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 4
2
- 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 1
3
p \end{array} \right)$$ Find

1. the unit vector in the direction of \(\overrightarrow { A B }\),
2. the value of the constant \(p\) for which angle \(B O C = 90 ^ { \circ }\). 3 The first three terms in the expansion of \(( 1 - 2 x ) ^ { 2 } ( 1 + a x ) ^ { 6 }\), in ascending powers of \(x\), are \(1 - x + b x ^ { 2 }\). Find the values of the constants \(a\) and \(b\). 4
3. Solve the equation \(\sin 2 x + 3 \cos 2 x = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
4. How many solutions has the equation \(\sin 2 x + 3 \cos 2 x = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 1080 ^ { \circ }\) ?

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 \(\overrightarrow { A B }\).

8 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { c } 3 p \\ 4 \\ p ^ { 2 } \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { c } - p \\ - 1 \\ p ^ { 2 } \end{array} \right)$$

1. Find the values of \(p\) for which angle \(A O B\) is \(90 ^ { \circ }\).
2. For the case where \(p = 3\), find the unit vector in the direction of \(\overrightarrow { B A }\).

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.

2 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r }

5
1
3 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 5
4
- 3 \end{array} \right) .$$ The point \(P\) lies on \(A B\) and is such that \(\overrightarrow { A P } = \frac { 1 } { 3 } \overrightarrow { A B }\).

1. Find the position vector of \(P\).
2. Find the distance \(O P\).
3. Determine whether \(O P\) is perpendicular to \(A B\). Justify your answer.
   5
4. Show that the equation \(\frac { 2 \sin \theta + \cos \theta } { \sin \theta + \cos \theta } = 2 \tan \theta\) may be expressed as \(\cos ^ { 2 } \theta = 2 \sin ^ { 2 } \theta\).
5. Hence solve the equation \(\frac { 2 \sin \theta + \cos \theta } { \sin \theta + \cos \theta } = 2 \tan \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

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

1. Find the value of \(k\) for which angle \(A O B\) is \(90 ^ { \circ }\).
2. Find the values of \(k\) for which the lengths of \(O A\) and \(O B\) are equal.
   The point \(C\) is such that \(\overrightarrow { A C } = 2 \overrightarrow { C B }\).
3. In the case where \(k = 4\), find the unit vector in the direction of \(\overrightarrow { O C }\).

5 Two vectors, \(\mathbf { u }\) and \(\mathbf { v }\), are such that $$\mathbf { u } = \left( \begin{array} { l } q
2

8
q - 1
q ^ { 2 } - 7 \end{array} \right)$$ where \(q\) is a constant.

1. Find the values of \(q\) for which \(\mathbf { u }\) is perpendicular to \(\mathbf { v }\).
2. Find the angle between \(\mathbf { u }\) and \(\mathbf { v }\) when \(q = 0\).
   6
3. The first and second terms of a geometric progression are \(p\) and \(2 p\) respectively, where \(p\) is a positive constant. The sum of the first \(n\) terms is greater than \(1000 p\). Show that \(2 ^ { n } > 1001\). [2]
4. In another case, \(p\) and \(2 p\) are the first and second terms respectively of an arithmetic progression. The \(n\)th term is 336 and the sum of the first \(n\) terms is 7224 . Write down two equations in \(n\) and \(p\) and hence find the values of \(n\) and \(p\).
   7 (a) Solve the equation \(3 \sin ^ { 2 } 2 \theta + 8 \cos 2 \theta = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
   (b) \includegraphics[max width=\textwidth, alt={}, center]{c8ac31bc-0f76-4d00-a28b-4a07758f9663-13_540_750_251_735} The diagram shows part of the graph of \(y = a + \tan b x\), where \(x\) is measured in radians and \(a\) and \(b\) are constants. The curve intersects the \(x\)-axis at \(\left( - \frac { 1 } { 6 } \pi , 0 \right)\) and the \(y\)-axis at \(( 0 , \sqrt { } 3 )\). Find the values of \(a\) and \(b\).
   8
5. Express \(x ^ { 2 } - 4 x + 7\) in the form \(( x + a ) ^ { 2 } + b\).
   The function f is defined by \(\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 7\) for \(x < k\), where \(k\) is a constant.
6. State the largest value of \(k\) for which f is a decreasing function.
   The value of \(k\) is now given to be 1 .
7. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
8. The function g is defined by \(\mathrm { g } ( x ) = \frac { 2 } { x - 1 }\) for \(x > 1\). Find an expression for \(\mathrm { gf } ( x )\) and state the range of gf.

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 \(O A\).

8 Relative to an origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } + 7 \mathbf { j } - p \mathbf { k }\) and the point \(B\) has position vector \(8 \mathbf { i } - \mathbf { j } - p \mathbf { k }\), where \(p\) is a constant.

1. Find \(\overrightarrow { O A } \cdot \overrightarrow { O B }\).
2. Hence show that there are no real values of \(p\) for which \(O A\) and \(O B\) are perpendicular to each other.
3. Find the values of \(p\) for which angle \(A O B = 60 ^ { \circ }\).

3 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 5 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 2 \mathbf { i } + 7 \mathbf { j } + p \mathbf { k }$$ where \(p\) is a constant.

1. Find the value of \(p\) for which angle \(A O B\) is \(90 ^ { \circ }\).
2. In the case where \(p = 4\), find the vector which has magnitude 28 and is in the same direction as \(\overrightarrow { A B }\).

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 \(C\).

7 Three points, \(O , A\) and \(B\), are such that \(\overrightarrow { O A } = \mathbf { i } + 3 \mathbf { j } + p \mathbf { k }\) and \(\overrightarrow { O B } = - 7 \mathbf { i } + ( 1 - p ) \mathbf { j } + p \mathbf { k }\), where \(p\) is a constant.

1. Find the values of \(p\) for which \(\overrightarrow { O A }\) is perpendicular to \(\overrightarrow { O B }\).
2. The magnitudes of \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) are \(a\) and \(b\) respectively. Find the value of \(p\) for which \(b ^ { 2 } = 2 a ^ { 2 }\).
3. Find the unit vector in the direction of \(\overrightarrow { A B }\) when \(p = - 8\).

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 \(O A\).

8

1. Relative to an origin \(O\), the position vectors of two points \(P\) and \(Q\) are \(\mathbf { p }\) and \(\mathbf { q }\) respectively. The point \(R\) is such that \(P Q R\) is a straight line with \(Q\) the mid-point of \(P R\). Find the position vector of \(R\) in terms of \(\mathbf { p }\) and \(\mathbf { q }\), simplifying your answer.
2. The vector \(6 \mathbf { i } + a \mathbf { j } + b \mathbf { k }\) has magnitude 21 and is perpendicular to \(3 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\). Find the possible values of \(a\) and \(b\), showing all necessary working.

2 Points \(A , B\) and \(C\) have position vectors \(- 5 \mathbf { i } - 10 \mathbf { j } + 12 \mathbf { k } , \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\) and \(3 \mathbf { i } + 6 \mathbf { j } + p \mathbf { k }\) respectively, where \(p\) is a constant.

1. Given that angle \(A B C = 90 ^ { \circ }\), find the value of \(p\).
2. Given instead that \(A B C\) is a straight line, find the value of \(p\).

6 Lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } + a \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 3 \mathbf { i } - \mathbf { k } + s ( 2 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } )$$ respectively, where \(t\) and \(s\) are parameters and \(a\) is a constant.

1. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular, find the value of \(a\).
2. Given instead that \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find
   (a) the value of \(a\),
   (b) the angle between the lines.

9 Two lines have equations $$\mathbf { r } = 3 \mathbf { i } + 5 \mathbf { j } - \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \text { and } \mathbf { r } = 4 \mathbf { i } + 10 \mathbf { j } + 19 \mathbf { k } + \mu ( \mathbf { i } - \mathbf { j } + \alpha \mathbf { k } ) ,$$ where \(\alpha\) is a constant.
Find the value of \(\alpha\) in each of the following cases.

1. The lines intersect at the point (7,7,1).
2. The angle between their directions is \(60 ^ { \circ }\).

4 The points A , B and C have coordinates \(( 1,3 , - 2 ) , ( - 1,2 , - 3 )\) and \(( 0 , - 8,1 )\) respectively.

1. Find the vectors \(\overrightarrow { \mathrm { AB } }\) and \(\overrightarrow { \mathrm { AC } }\).
2. Show that the vector \(2 \mathbf { i } - \mathbf { j } - 3 \mathbf { k }\) is perpendicular to the plane ABC . Hence find the equation of the plane ABC .

2 The two vectors \(\mathbf { a }\) and \(\mathbf { b }\) are such that \(\mathbf { a } \cdot \mathbf { b } = 0\) State the angle between the vectors \(\mathbf { a }\) and \(\mathbf { b }\) Circle your answer.
[0pt] [1 mark] \(0 ^ { \circ } 45 ^ { \circ } 90 ^ { \circ } 180 ^ { \circ }\)

1. All units in this question are in metres.

A lawn is modelled as a plane that contains the points \(L ( - 2 , - 3 , - 1 ) , M ( 6 , - 2,0 )\) and \(N ( 2,0,0 )\), relative to a fixed origin \(O\).

1. Determine a vector equation of the plane that models the lawn, giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\)
2. 1. Show that, according to the model, the lawn is perpendicular to the vector \(\left( \begin{array} { c } 1 \\ 2 \\ - 10 \end{array} \right)\)
   2. Hence determine a Cartesian equation of the plane that models the lawn. There are two posts set in the lawn.
      There is a washing line between the two posts.
      The washing line is modelled as a straight line through points at the top of each post with coordinates \(P ( - 10,8,2 )\) and \(Q ( 6,4,3 )\).
3. Determine a vector equation of the line that models the washing line.
4. State a limitation of one of the models. The point \(R ( 2,5,2.75 )\) lies on the washing line.
5. Determine, according to the model, the shortest distance from the point \(R\) to the lawn, giving your answer to the nearest cm. Given that the shortest distance from the point \(R\) to the lawn is actually 1.5 m ,
6. use your answer to part (e) to evaluate the model, explaining your reasoning.