Perpendicular distance point to plane

7 The straight line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k } )\). The plane \(p\) passes through the point \(( 4 , - 1,2 )\) and is perpendicular to \(l\).

1. Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the perpendicular distance from the origin to \(p\).
3. A second plane \(q\) is parallel to \(p\) and the perpendicular distance between \(p\) and \(q\) is 14 units. Find the possible equations of \(q\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = a \mathbf { i } + 9 \mathbf { j } + 13 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 3 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } )$$ respectively. It is given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.

1. Find the value of the constant \(a\).
   The point \(P\) has position vector \(3 \mathbf { i } + \mathbf { j } + 6 \mathbf { k }\).
2. Find the perpendicular distance from \(P\) to the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
3. Find the perpendicular distance from \(P\) to \(l _ { 2 }\).

6 The equation of the plane \(\Pi\) is \(\mathbf { r } = \left( \begin{array} { r } - 1 \\ 2 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { l } 4 \\ 4 \\ 3 \end{array} \right) + \mu \left( \begin{array} { r } - 2 \\ 3 \\ 1 \end{array} \right)\).

1. Find the acute angle between \(\Pi\) and the plane with equation \(\mathbf { r } . \left( \begin{array} { l } 2 \\ 0 \\ 3 \end{array} \right) = 4\). The point \(A\) has coordinates ( \(9 , - 7,20\) ).
   The point \(F\) is the point of intersection between \(\Pi\) and the perpendicular from \(A\) to \(\Pi\).
2. Determine the coordinates of \(F\).

8 The line \(l\) has equation \(\mathbf { r } = \lambda \mathbf { d }\) and the plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } . \mathbf { n } = 35\), where $$\mathbf { d } = \left( \begin{array} { r } 2 \\ - 1 \\ 2 \end{array} \right) \quad \text { and } \quad \mathbf { n } = \left( \begin{array} { r } 6 \\ - 2 \\ 3 \end{array} \right) .$$

1. (a) Determine the exact value of \(\cos \theta\), where \(\theta\) is the angle between \(\mathbf { d }\) and \(\mathbf { n }\).
   (b) Determine the position vector of the point of intersection of \(l\) and \(\Pi _ { 1 }\).
   (c) Determine the shortest distance from \(O\) to \(\Pi _ { 1 }\).
2. The plane \(\Pi _ { 2 }\) has cartesian equation \(12 x - 4 y + 6 z + 21 = 0\). Determine the distance between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

The plane \(P\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 2 \\ 2 \end{pmatrix}$$

1. Find a vector perpendicular to the plane \(P\). [2] 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. [4]
3. Find the perpendicular distance from \(A\) to the plane \(P\). [4]

The line \(l\) passes through the point \(P(2, 1, 3)\) and is perpendicular to the plane \(\Pi\) whose vector equation is $$\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) = 3$$ Find

1. a vector equation of the line \(l\), [2]
2. the position vector of the point where \(l\) meets \(\Pi\). [4]
3. Hence find the perpendicular distance of \(P\) from \(\Pi\). [2]

The plane \(\Pi\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 3 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that vector \(\mathbf{2i + 3j - 4k}\) is perpendicular to \(\Pi\). [2]
2. Hence find a Cartesian equation of \(\Pi\). [2]

The line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ -5 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ 6 \\ -3 \end{pmatrix}$$ where \(t\) is a scalar parameter. The point \(A\) lies on \(l\). Given that the shortest distance between \(A\) and \(\Pi\) is \(2\sqrt{29}\)

1. determine the possible coordinates of \(A\). [4]

The plane \(\Pi\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 3 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that vector \(2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) is perpendicular to \(\Pi\). [2]
2. Hence find a Cartesian equation of \(\Pi\). [2]

The line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ -5 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ 6 \\ -3 \end{pmatrix}$$ where \(t\) is a scalar parameter. The point \(A\) lies on \(l\). Given that the shortest distance between \(A\) and \(\Pi\) is \(2\sqrt{29}\)

1. determine the possible coordinates of \(A\). [4]

The equation of a plane, \(\Pi\), is $$\Pi: \mathbf{r} = \begin{pmatrix} 2 \\ -3 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}.$$

1. Find a vector which is perpendicular to \(\Pi\). [2]
2. Hence find an equation for \(\Pi\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [2]
3. Find in the form \(\sqrt{q}\) the shortest distance between \(\Pi\) and the origin, where \(q\) is a rational number. [2]