Line intersection with plane

9 A line has Cartesian equations \(x - p = \frac { y + 2 } { q } = 3 - z\) and a plane has
equation r. \(\left[ \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right] = - 3\) 9

1. In the case where the plane fully contains the line, find the values of \(p\) and \(q\).
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   9
2. In the case where the line intersects the plane at a single point, find the range of values of \(p\) and \(q\).
   [0pt] [3 marks]
   9
3. In the case where the angle \(\theta\) between the line and the plane satisfies \(\sin \theta = \frac { 1 } { \sqrt { 6 } }\) and the line intersects the plane at \(z = 0\) 9
4. 1. Find the value of \(q\).
      [0pt] [4 marks]
      9
5. (ii) Find the value of \(p\).

4 The equations of two intersecting lines \(l _ { 1 }\) and \(l _ { 2 }\) are
\(l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ a \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 1 \\ - 3 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 7 \\ 9 \\ - 2 \end{array} \right) + \mu \left( \begin{array} { r } - 1 \\ 1 \\ 2 \end{array} \right)\)
where \(a\) is a constant.
The equation of the plane \(\Pi\) is
r. \(\left( \begin{array} { l } 1 \\ 5 \\ 3 \end{array} \right) = - 14\).
\(l _ { 1 }\) and \(\Pi\) intersect at \(Q\).
\(l _ { 2 }\) and \(\Pi\) intersect at \(R\).

1. Verify that the coordinates of \(R\) are (13, 3, -14).
2. Determine the exact value of the length of \(Q R\).

3 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ - 3 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ 2 \\ - 2 \end{array} \right)\).
The plane \(\Pi\) has equation \(\mathbf { r } \cdot \left( \begin{array} { r } 2 \\ - 5 \\ - 3 \end{array} \right) = 4\).

1. Find the position vector of the point of intersection of \(l _ { 1 }\) and \(\Pi\).
2. Find the acute angle between \(l _ { 1 }\) and \(\Pi\).
   \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 1\).
   \(l _ { 2 }\) is the line with the following properties.
   • \(l _ { 2 }\) passes through \(A\)
   • \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\)
   • \(l _ { 2 }\) is parallel to \(\Pi\)
   • Find, in vector form, the equation of \(l _ { 2 }\).

3. The vector equation of the line \(L\) is given by $$\mathbf { r } = - \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } + \lambda ( 4 \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } ) .$$ The Cartesian equation of the plane \(\Pi\) is given by $$3 x + 8 y - 9 z = 0$$ Find the Cartesian coordinates of the point of intersection of \(L\) and \(\Pi\).

5. The points \(A\) and \(B\) have coordinates \(( 3,4 , - 2 )\) and \(( - 2,0,7 )\) respectively. The equation of the plane \(\Pi\) is given by \(2 x + 3 y + 3 z = 27\).

1. Show that the vector equation of the line \(A B\) may be expressed in the form $$\mathbf { r } = ( 3 - 5 \lambda ) \mathbf { i } + ( 4 - 4 \lambda ) \mathbf { j } + ( - 2 + 9 \lambda ) \mathbf { k }$$
2. Find the coordinates of the point of intersection of the line \(A B\) and the plane \(\Pi\).

8. The line \(L\) passes through the points \(\mathrm { A } ( 1,2,3 )\) and \(\mathrm { B } ( 2,3,1 )\).

1. 1. Find the vector \(\mathbf { A B }\).
   2. Write down the vector equation of the line \(L\).
2. The plane \(\Pi\) has equation \(x + 3 y - 2 z = 5\).
   1. Find the coordinates of the point of intersection of \(L\) and \(\Pi\).
   2. Find the acute angle between \(L\) and \(\Pi\).

1. The plane \(\Pi _ { 1 }\) has equation

$$\mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } ) + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Find a Cartesian equation for \(\Pi _ { 1 }\) The line \(l\) has equation $$\frac { x - 1 } { 5 } = \frac { y - 3 } { - 3 } = \frac { z + 2 } { 4 }$$
2. Find the coordinates of the point of intersection of \(l\) with \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r . } ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 5$$
3. Find, to the nearest degree, the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)

1 Plane \(\Pi\) has equation \(3 x - y + 2 z = 33\). Line \(l\) has the following vector equation. $$l : \quad \mathbf { r } = \left( \begin{array} { l } 1
0
5 \end{array} \right) + \lambda \left( \begin{array} { l }

2 The equations of two intersecting lines \(l _ { 1 }\) and \(l _ { 2 }\) are
\(l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ a \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 1 \\ - 3 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 7 \\ 9 \\ - 2 \end{array} \right) + \mu \left( \begin{array} { r } - 1 \\ 1 \\ 2 \end{array} \right)\)
where \(a\) is a constant.
The equation of the plane \(\Pi\) is
r. \(\left( \begin{array} { l } 1 \\ 5 \\ 3 \end{array} \right) = - 14\).
\(l _ { 1 }\) and \(\Pi\) intersect at \(Q\).
\(\zeta _ { 2 }\) and \(\Pi\) intersect at \(R\).

1. Verify that the coordinates of \(R\) are \(( 13,3 , - 14 )\).
2. Determine the exact value of the length of \(Q R\).