Perpendicular distance from point to line

8 The point \(P\) has coordinates \(( - 1,4,11 )\) and the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ 3 \\ - 4 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right)\).

1. Find the perpendicular distance from \(P\) to \(l\).
2. Find the equation of the plane which contains \(P\) and \(l\), giving your answer in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers.

11 \includegraphics[max width=\textwidth, alt={}, center]{ce3c4a9c-bf83-4d28-96e2-ef31c3673dea-16_593_780_264_685} In the diagram, \(O A B C D E F G\) is a cuboid in which \(O A = 3\) units, \(O C = 2\) units and \(O D = 2\) units. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O D\) and \(O C\) respectively. \(M\) is the midpoint of \(E F\).

1. Find the position vector of \(M\).
   The position vector of \(P\) is \(\mathbf { i } + \mathbf { j } + 2 \mathbf { k }\).
2. Calculate angle PAM.
3. Find the exact length of the perpendicular from \(P\) to the line passing through \(O\) and \(M\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1. Relative to a fixed origin \(O\), the line \(l\) has equation

$$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 10 \\ - 9 \end{array} \right) + \lambda \left( \begin{array} { l } 4 \\ 4 \\ 2 \end{array} \right) \quad \text { where } \lambda \text { is a scalar parameter }$$ Given that \(\overrightarrow { O A }\) is a unit vector parallel to \(l\),

1. find \(\overrightarrow { O A }\) The point \(X\) lies on \(l\).
   Given that \(X\) is the point on \(l\) that is closest to the origin,
2. find the coordinates of \(X\). The points \(O , X\) and \(A\) form the triangle \(O X A\).
3. Find the exact area of triangle \(O X A\).

3 Find the perpendicular distance from the point with position vector \(12 \mathbf { i } + 5 \mathbf { j } + 3 \mathbf { k }\) to the line with equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } + t ( 8 \mathbf { i } + 3 \mathbf { j } - 6 \mathbf { k } )\).

4 The line \(l\) has equations \(\frac { x - 1 } { 2 } = \frac { y - 1 } { 3 } = \frac { z + 1 } { 2 }\) and the point \(A\) is ( \(7,3,7\) ). \(M\) is the point where the perpendicular from \(A\) meets \(l\).

1. Find, in either order, the coordinates of \(M\) and the perpendicular distance from \(A\) to \(l\).
2. Find the coordinates of the point \(B\) on \(A M\) such that \(\overrightarrow { A B } = 3 \overrightarrow { B M }\).

2
The position vector of point \(A\) is \(\mathbf { a } = - 9 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k }\).
The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf { a }\).

1. Determine the shortest distance between the origin, \(O\), and \(l\). \(l\) is also perpendicular to the vector \(\mathbf { b }\) where \(\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }\).
2. Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).
3. Write down an equation of \(l\) in vector form. \(P\) is a point on \(l\) such that \(P A = 2 O A\).
4. Find angle \(P O A\) giving your answer to 3 significant figures. \(C\) is a point whose position vector, \(\mathbf { c }\), is given by \(\mathbf { c } = p \mathbf { a }\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf { r } = \mathbf { c } + \mu \mathbf { b }\). The point with position vector \(9 \mathbf { i } + 8 \mathbf { j } - 12 \mathbf { k }\) lies on \(m\).
5. Find the value of \(p\). \section*{In this question you must show detailed reasoning.} You are given that \(\alpha , \beta\) and \(\gamma\) are the roots of the equation \(5 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0\).
6. Find the value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
7. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\) giving your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) where \(a , b , c\) and \(d\) are integers.