Perpendicularity conditions

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 \(\overrightarrow { B A }\).

7 Given that \(\mathbf { a } = \left( \begin{array} { r } 2 \\ - 2 \\ 1 \end{array} \right) , \mathbf { b } = \left( \begin{array} { l } 2 \\ 6 \\ 3 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { c } p \\ p \\ p + 1 \end{array} \right)\), find

1. the angle between the directions of \(\mathbf { a }\) and \(\mathbf { b }\),
2. the value of \(p\) for which \(\mathbf { b }\) and \(\mathbf { c }\) are perpendicular.

9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ 3 \\ - 6 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 0 \\ - 6 \\ 8 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } - 2 \\ 5 \\ - 2 \end{array} \right)$$

1. Find angle \(A O B\).
2. Find the vector which is in the same direction as \(\overrightarrow { A C }\) and has magnitude 30 .
3. Find the value of the constant \(p\) for which \(\overrightarrow { O A } + p \overrightarrow { O B }\) is perpendicular to \(\overrightarrow { O C }\).

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

1. Use a scalar product to find angle \(A O B\).
2. Find the vector which is in the same direction as \(\overrightarrow { A C }\) and of magnitude 15 units.
3. Find the value of the constant \(p\) for which \(p \overrightarrow { O A } + \overrightarrow { O C }\) is perpendicular to \(\overrightarrow { O B }\).

7
2
- 5 \end{array} \right)$$ Given that $$\overrightarrow { A B } = \left( \begin{array} { r } - 2
4
3 \end{array} \right)$$

1. find the coordinates of the point \(B\). The point \(C\) has position vector $$\overrightarrow { O C } = \left( \begin{array} { r } a
   5
   - 1 \end{array} \right)$$ where \(a\) is a constant.\\ Given that \(\overrightarrow { O C }\) is perpendicular to \(\overrightarrow { B C }\)
2. find the possible values of \(a\).
   1. The curve \(C\) is defined by the equation
   $$8 x ^ { 3 } - 3 y ^ { 2 } + 2 x y = 9$$ Find an equation of the normal to \(C\) at the point ( 2,5 ), giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e583bf92-d6a9-4f1a-b3c8-372afa6e0a0e-08_558_542_258_749} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of a segment \(P Q R P\) of a circle with centre \(O\) and radius 5 cm .\\ Given that
   • angle \(P O R\) is \(\theta\) radians
   • \(\theta\) is increasing, from 0 to \(\pi\), at a constant rate of 0.1 radians per second
   • the area of the segment \(P Q R P\) is \(A \mathrm {~cm} ^ { 2 }\)
   • show that
   $$\frac { \mathrm { d } A } { \mathrm {~d} \theta } = K ( 1 - \cos \theta )$$ where \(K\) is a constant to be found.
3. Find, in \(\mathrm { cm } ^ { 2 } \mathrm {~s} ^ { - 1 }\), the rate of increase of the area of the segment when \(\theta = \frac { \pi } { 3 }\) 5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e583bf92-d6a9-4f1a-b3c8-372afa6e0a0e-10_803_1086_248_493} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of the curve defined by the parametric equations $$x = t ^ { 2 } + 2 t \quad y = \frac { 2 } { t ( 3 - t ) } \quad a \leqslant t \leqslant b$$ where \(a\) and \(b\) are constants.\\ The ends of the curve lie on the line with equation \(y = 1\)
4. Find the value of \(a\) and the value of \(b\) The region \(R\), shown shaded in Figure 2, is bounded by the curve and the line with equation \(y = 1\)
5. Show that the area of region \(R\) is given by $$M - k \int _ { a } ^ { b } \frac { t + 1 } { t ( 3 - t ) } \mathrm { d } t$$ where \(M\) and \(k\) are constants to be found.
6. 1. Write \(\frac { t + 1 } { t ( 3 - t ) }\) in partial fractions.
   2. Use algebraic integration to find the exact area of \(R\), giving your answer in simplest form.
      1. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation
   $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } + \lambda ( 8 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } )$$ where \(\lambda\) is a scalar parameter.\\ The point \(A\) lies on \(l _ { 1 }\)\\ Given that \(| \overrightarrow { O A } | = 5 \sqrt { 10 }\)
7. show that at \(A\) the parameter \(\lambda\) satisfies $$81 \lambda ^ { 2 } + 52 \lambda - 220 = 0$$ Hence
8. 1. show that one possible position vector for \(A\) is \(- 15 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\)
   2. find the other possible position vector for \(A\). The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through \(O\).\\ Given that
      • \(\overrightarrow { O A } = - 15 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\)
9. point \(B\) lies on \(l _ { 2 }\) where \(| \overrightarrow { O B } | = 4 \sqrt { 10 }\)
10. find the area of triangle \(O A B\), giving your answer to one decimal place.
1. The current, \(x\) amps, at time \(t\) seconds after a switch is closed in a particular electric circuit is modelled by the equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = k - 3 x$$ where \(k\) is a constant.
Initially there is zero current in the circuit.
11. Solve the differential equation to find an equation, in terms of \(k\), for the current in the circuit at time \(t\) seconds.
    Give your answer in the form \(x = \mathrm { f } ( t )\). Given that in the long term the current in the circuit approaches 7 amps,
12. find the value of \(k\).
13. Hence find the time in seconds it takes for the current to reach 5 amps, giving your answer to 2 significant figures.

1. Points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\), respectively, relative to an origin \(O\), and are such that \(O A B\) is a triangle with \(O A = a\) and \(O B = b\).

The point \(C\), with position vector \(\mathbf { c }\), lies on the line through \(O\) that bisects the angle \(A O B\).

1. Prove that the vector \(b \mathbf { a } - a \mathbf { b }\) is perpendicular to \(\mathbf { c }\). The point \(D\), with position vector \(\mathbf { d }\), lies on the line \(A B\) between \(A\) and \(B\).
2. Explain why \(\mathbf { d }\) can be expressed in the form \(\mathbf { d } = ( 1 - \lambda ) \mathbf { a } + \lambda \mathbf { b }\) for some scalar \(\lambda\) with \(0 < \lambda < 1\)
3. Given that \(D\) is also on the line \(O C\), find an expression for \(\lambda\) in terms of \(a\) and \(b\) only and hence show that $$D A : D B = O A : O B$$

8 The points \(A\) and \(B\) have position vectors relative to the origin \(O\) given by $$\overrightarrow { O A } = \left( \begin{array} { c } 3 \sin \alpha \\ 2 \cos \alpha \\ - 1 \end{array} \right) \text { and } \overrightarrow { O B } = \left( \begin{array} { c } 2 \cos \alpha \\ 4 \sin \alpha \\ 3 \end{array} \right)$$ where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). It is given that \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) are perpendicular.

1. Calculate the two possible values of \(\alpha\).
2. Calculate the area of triangle \(O A B\) for the smaller value of \(\alpha\) from part (i).

1

1. Determine whether the point \(( 19 , - 12,17 )\) lies on the line \(\mathbf { r } = \left( \begin{array} { r } 4 \\ - 2 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right)\). Vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by \(\mathbf { a } = \left( \begin{array} { r } 1 \\ - 2 \\ 2 \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { r } - 3 \\ 6 \\ 2 \end{array} \right)\).
2. 1. Find, in degrees, the angle between \(\mathbf { a }\) and \(\mathbf { b }\).
   2. Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).

8 The line segment \(A B\) is a diameter of a sphere, \(S\). The point \(C\) is any point on the surface of \(S\).

1. Explain why \(\overrightarrow { \mathrm { AC } } \cdot \overrightarrow { \mathrm { BC } } = 0\) for all possible positions of \(C\). You are now given that \(A\) is the point ( \(11,12 , - 14\) ) and \(B\) is the point ( \(9,13,6\) ).
2. Given that the coordinates of \(C\) have the form ( \(2 p , p , 1\) ), where \(p\) is a constant, determine the coordinates of the possible positions of \(C\). \section*{END OF QUESTION PAPER}

7. The vector equations of the lines \(L _ { 1 } , L _ { 2 } , L _ { 3 }\) are given by $$\begin{aligned} & \mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( 2 \mathbf { i } + n \mathbf { j } + \mathbf { k } ) \\ & \mathbf { r } = 5 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } ) \\ & \mathbf { r } = 6 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } + v ( p \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) \end{aligned}$$ respectively, where \(n\) and \(p\) are constants.
The line \(L _ { 1 }\) is perpendicular to the line \(L _ { 2 }\). The line \(L _ { 1 }\) is also perpendicular to the line \(L _ { 3 }\).

1. Show that the value of \(n\) is - 3 and find the value of \(p\).
2. Find the acute angle between the lines \(L _ { 2 }\) and \(L _ { 3 }\).