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 \(\overrightarrow { O B }\).

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 \(L _ { 3 }\).