Angle between line and plane

10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + \lambda ( a \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\), where \(a\) is a constant. The plane \(p\) has equation \(x + 2 y + 2 z = 6\). Find the value or values of \(a\) in each of the following cases.

1. The line \(l\) is parallel to the plane \(p\).
2. The line \(l\) intersects the line passing through the points with position vectors \(3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and \(\mathbf { i } + \mathbf { j } - \mathbf { k }\).
3. The acute angle between the line \(l\) and the plane \(p\) is \(\tan ^ { - 1 } 2\).

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

$$\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 3 \\ - 2 \end{array} \right) + \mu \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Determine a vector perpendicular to \(\Pi\) The line \(l\) meets \(\Pi\) at the point ( \(1,2,3\) ) and passes through the point ( \(1,0,1\) )
2. Determine the size of the acute angle between \(\Pi\) and \(l\) Give your answer to the nearest degree.
3. Determine the shortest distance between \(\Pi\) and the point \(( 6 , - 3 , - 6 )\)

1. The plane \(P\) has equation

$$\mathbf { r } = \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 2 \\ - 1 \end{array} \right) + \mu \left( \begin{array} { l } 3 \\ 2 \\ 2 \end{array} \right)$$

1. Find a vector perpendicular to the plane \(P\). The line \(l\) passes through the point \(A ( 1,3,3 )\) and meets \(P\) at \(( 3,1,2 )\). The acute angle between the plane \(P\) and the line \(l\) is \(\alpha\).
2. Find \(\alpha\) to the nearest degree.
3. Find the perpendicular distance from \(A\) to the plane \(P\).

1 A line \(l\) has equation \(\frac { x - 1 } { 5 } = \frac { y - 6 } { 6 } = \frac { z + 3 } { - 7 }\) and a plane \(p\) has equation \(x + 2 y - z = 40\).

1. Find the acute angle between \(l\) and \(p\).
2. Find the perpendicular distance from the point \(( 1,6 , - 3 )\) to \(p\).

4 Determine the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } - \sqrt { 3 } \\ 1 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { c } 1 \\ 2 \sqrt { 3 } \\ - \sqrt { 3 } \end{array} \right)\) and the \(y\)-axis.

6 The equation of a plane \(\Pi\) is \(x - 2 y - z = 30\).

1. Find the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } 3 \\ 2 \\ - 5 \end{array} \right) + \lambda \left( \begin{array} { r } - 5 \\ 3 \\ 2 \end{array} \right)\) and \(\Pi\).
2. Determine the geometrical relationship between the line \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 4 \\ 2 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 1 \\ 5 \end{array} \right)\) and \(\Pi\).

2 In this question you must show detailed reasoning.
Find the angle between the vector \(3 i + 2 j + \mathbf { k }\) and the plane \(- x + 3 y + 2 z = 8\).

6 The equation of a plane \(\Pi\) is \(x - 2 y - z = 30\).

1. Find the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } 3 \\ 2 \\ - 5 \end{array} \right) + \lambda \left( \begin{array} { r } - 5 \\ 3 \\ 2 \end{array} \right)\) and \(\Pi\).
2. Determine the geometrical relationship between the line \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 4 \\ 2 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 1 \\ 5 \end{array} \right)\) and \(\Pi\).

5 A plane has equation r. \(\left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right] = 7\)
A line has equation \(\mathbf { r } = \left[ \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right] + \mu \left[ \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right]\)
Calculate the acute angle between the line and the plane.
Give your answer to the nearest \(0.1 ^ { \circ }\)
\includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-05_2491_1716_219_153}