Reflection in plane

9 The equations of two straight lines \(l _ { 1 }\) and \(l _ { 2 }\) are $$l _ { 1 } : \quad \mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad l _ { 2 } : \quad \mathbf { r } = - \mathbf { i } - \mathbf { j } - \mathbf { k } + \mu ( 3 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } ) ,$$ where \(a\) is a constant.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.

1. Show that \(a = 4\).
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) also intersect.
2. Find the position vector of the point of intersection.
   The point \(A\) has position vector \(- 5 \mathbf { i } + \mathbf { j } - 9 \mathbf { k }\).
3. Show that \(A\) lies on \(l _ { 1 }\).
   The point \(B\) is the image of \(A\) after a reflection in the line \(l _ { 2 }\).
4. Find the position vector of \(B\).

1. Relative to a fixed origin \(O\),

• the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } + \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\)
• the point \(P\) has position vector \(2 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\)

The straight line \(l\) passes through \(A\) and \(B\).

1. Find a vector equation for \(l\). The point \(C\) lies on \(l\) so that \(P C\) is perpendicular to \(l\).
2. Find the coordinates of \(C\). The point \(P ^ { \prime }\) is the reflection of \(P\) in the line \(l\).
3. Find the coordinates of \(P ^ { \prime }\)
4. Hence find \(\left| \overrightarrow { P P ^ { \prime } } \right|\), giving your answer as a simplified surd.

8. The line \(l\) has equation $$\mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) , \text { where } \lambda \text { is a scalar parameter, }$$ and the plane \(\Pi\) has equation $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 19$$

1. Find the coordinates of the point of intersection of \(l\) and \(\Pi\). The perpendicular to \(\Pi\) from the point \(A ( 2,1 , - 2 )\) meets \(\Pi\) at the point \(B\).
2. Verify that the coordinates of \(B\) are \(( 4,3 , - 6 )\). The point \(A ( 2,1 , - 2 )\) is reflected in the plane \(\Pi\) to give the image point \(A ^ { \prime }\).
3. Find the coordinates of the point \(A ^ { \prime }\).
4. Find an equation for the line obtained by reflecting the line \(l\) in the plane \(\Pi\), giving your answer in the form $$\mathbf { r } \times \mathbf { a } = \mathbf { b } ,$$ where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be found.

1. The plane \(\Pi\) has equation

$$3 x + 4 y - z = 17$$ The line \(l _ { 1 }\) is perpendicular to \(\Pi\) and passes through the point \(P ( - 4 , - 5,3 )\)
The line \(l _ { 1 }\) intersects \(\Pi\) at the point \(Q\)

1. Determine the coordinates of \(Q\) Given that the point \(R ( - 1,6,4 )\) lies on \(\Pi\)
2. determine a Cartesian equation of the plane containing \(P Q R\) The line \(l _ { 2 }\) passes through \(P\) and \(R\)
   The line \(l _ { 3 }\) is the reflection of \(l _ { 2 }\) in \(\Pi\)
3. Determine a vector equation for \(l _ { 3 }\)

1 With respect to cartesian coordinates Oxyz, a laser beam ABC is fired from the point \(\mathrm { A } ( 1,2,4 )\), and is reflected at point B off the plane with equation \(x + 2 y - 3 z = 0\), as shown in Fig. 8. \(\mathrm { A } ^ { \prime }\) is the point \(( 2,4,1 )\), and \(M\) is the midpoint of \(\mathrm { AA } ^ { \prime }\). \begin{figure}[h] \end{figure}

1. Show that \(\mathrm { AA } ^ { \prime }\) is perpendicular to the plane \(x + 2 y - 3 z = 0\), and that M lies in the plane. The vector equation of the line AB is \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right)\).
2. Find the coordinates of B , and a vector equation of the line \(\mathrm { A } ^ { \prime } \mathrm { B }\).
3. Given that \(\mathrm { A } ^ { \prime } \mathrm { BC }\) is a straight line, find the angle \(\theta\).
4. Find the coordinates of the point where BC crosses the Oxz plane (the plane containing the \(x\) - and \(z\)-axes)

5 The line \(l _ { 1 }\) has equation \(\frac { x + 5 } { 1 } = \frac { y + 4 } { - 3 } = \frac { z - 3 } { 5 }\) The plane \(\Pi _ { 1 }\) has equation \(2 x + 3 y - 2 z = 6\)

1. Find the point of intersection of \(l _ { 1 }\) and \(\Pi _ { 1 }\) The line \(l _ { 2 }\) is the reflection of the line \(l _ { 1 }\) in the plane \(\Pi _ { 1 }\)
2. Show that a vector equation for the line \(l _ { 2 }\) is $$\mathbf { r } = \left( \begin{array} { r } - 7 \\ 2 \\ - 7 \end{array} \right) + \mu \left( \begin{array} { c } 10 \\ 6 \\ 2 \end{array} \right)$$ where \(\mu\) is a scalar parameter. The plane \(\Pi _ { 2 }\) contains the line \(l _ { 1 }\) and the line \(l _ { 2 }\)
3. Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) The plane \(\Pi _ { 3 }\) has equation r. \(\left( \begin{array} { l } 1 \\ 1 \\ a \end{array} \right) = b\) where \(a\) and \(b\) are constants.
   Given that the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) form a sheaf,
4. determine the value of \(a\) and the value of \(b\).

10 In this question all measurements are in centimetres. A small, thin laser pen is set up with one end at \(A ( 7,2 , - 3 )\) and the other end at \(B ( 9 , - 3 , - 2 )\) A laser beam travels from \(A\) to \(B\) and continues in a straight line towards a large thin sheet of glass. The sheet of glass lies within a plane \(\Pi _ { 1 }\) which is modelled by the equation $$4 x + p y + 5 z = 9$$ where \(p\) is an integer.
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1. The laser beam hits \(\Pi _ { 1 }\) at an acute angle \(\alpha\), where \(\sin \alpha = \frac { \sqrt { 15 } } { 75 }\)
   Find the value of \(p\)
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2. A second large sheet of glass lies on the other side of \(\Pi _ { 1 }\) This second sheet lies within a plane \(\Pi _ { 2 }\) which is modelled by the equation $$4 x + p y + 5 z = - 5$$ Calculate the distance between the sheets of glass.
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3. The point \(A ( 7,2 , - 3 )\) is reflected in \(\Pi _ { 1 }\)
   Find the coordinates of the image of \(A\) after reflection in \(\Pi _ { 1 }\)