Perpendicular distance from point to plane

1. The point \(P\) has coordinates \(( 1,2,1 )\)

The line \(l\) has Cartesian equation $$\frac { x - 3 } { 5 } = \frac { y + 1 } { 3 } = \frac { z + 5 } { - 8 }$$ The plane \(\Pi _ { 1 }\) contains the point \(P\) and the line \(l\).

1. Show that a Cartesian equation for \(\Pi _ { 1 }\) is $$6 x - 2 y + 3 z = 5$$ The point \(Q\) has coordinates \(( 2 , k , - 7 )\), where \(k\) is a constant.
2. Show that the shortest distance between \(\Pi _ { 1 }\) and \(Q\) is $$\frac { 2 } { 7 } | k + 7 |$$ The plane \(\Pi _ { 2 }\) has Cartesian equation \(8 x - 4 y + z = - 3\) Given that the shortest distance between \(\Pi _ { 1 }\) and \(Q\) is the same as the shortest distance between \(\Pi _ { 2 }\) and \(Q\),
3. determine the possible values of \(k\).
   

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

$$\mathbf { r } . ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) = 5$$

1. Find the perpendicular distance from the point \(( 6,2,12 )\) to the plane \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation $$\mathbf { r } = \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } ) + \mu ( \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) , \text { where } \lambda \text { and } \mu \text { are scalar parameters. }$$
2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer to the nearest degree.
3. Find an equation of the line of intersection of the two planes in the form \(\mathbf { r } \times \mathbf { a } = \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors.

The points \(A , B\) and \(C\) have position vectors \(\mathbf { i } , 2 \mathbf { j }\) and \(4 \mathbf { k }\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(A B C\). The point \(P\) on the line-segment \(O N\) is such that \(O P = \frac { 3 } { 4 } O N\). The line \(A P\) meets the plane \(O B C\) at \(Q\).

1. Find a vector perpendicular to the plane \(A B C\) and show that the length of \(O N\) is \(\frac { 4 } { \sqrt { } ( 21 ) }\).
2. Find the position vector of the point \(Q\).
3. Show that the acute angle between the planes \(A B C\) and \(A B Q\) is \(\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\).

The points \(A , B\) and \(C\) have position vectors \(\mathbf { i } , 2 \mathbf { j }\) and \(4 \mathbf { k }\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(A B C\). The point \(P\) on the line-segment \(O N\) is such that \(O P = \frac { 3 } { 4 } O N\). The line \(A P\) meets the plane \(O B C\) at \(Q\). Find a vector perpendicular to the plane \(A B C\) and show that the length of \(O N\) is \(\frac { 4 } { \sqrt { } ( 21 ) }\). Find the position vector of the point \(Q\). Show that the acute angle between the planes \(A B C\) and \(A B Q\) is \(\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\).

11 The plane \(\Pi\) has equation \(2 x - y + 2 z = 4\). The point \(P\) has coordinates \(( 8,4,5 )\).

1. Calculate the shortest distance from P to \(\Pi\). The line \(L\) has equation \(\frac { x - 2 } { 3 } = \frac { y } { 2 } = \frac { z + 3 } { 4 }\).
2. Verify that P lies on L .
3. Find the coordinates of the point of intersection of L and \(\Pi\).
4. Determine the acute angle between L and \(\Pi\).
5. Use the results of parts (b), (c) and (d) to verify your answer to part (a).

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

$$\mathbf { r } \cdot ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) = 5$$

1. Find the perpendicular distance from the point \(( 6,2,12 )\) to the plane \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation $$\mathbf { r } = \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } ) + \mu ( \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.
2. Show that the vector \(- \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\) is perpendicular to \(\Pi _ { 2 }\)
3. Show that the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) is \(52 ^ { \circ }\) to the nearest degree.

9 The position vectors of the points \(A , B\) and \(C\) are $$\begin{aligned} & \mathbf { a } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } \\ & \mathbf { b } = - \mathbf { i } - 8 \mathbf { j } + 2 \mathbf { k } \\ & \mathbf { c } = - 2 \mathbf { j } \end{aligned}$$ respectively.
9

1. Find the area of the triangle \(A B C\) 9
2. The points \(A , B\) and \(C\) all lie in the plane \(\Pi\) Find an equation of the plane \(\Pi\), in the form \(\mathbf { r } \cdot \mathbf { n } = d\) \(\mathbf { 9 ( c ) } \quad\) The point \(P\) has position vector \(\mathbf { p } = \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\) Find the exact distance of \(P\) from \(\Pi\)

7 The points \(A , B\) and \(C\) have coordinates \(A ( 4,5,2 ) , B ( - 3,2 , - 4 )\) and \(C ( 2,6,1 )\) 7

1. Use a vector product to show that the area of triangle \(A B C\) is \(\frac { 5 \sqrt { 11 } } { 2 }\) [0pt] [4 marks]
   7
2. The points \(A , B\) and \(C\) lie in a plane.
   Find a vector equation of the plane in the form r.n \(= k\) 7
3. Hence find the exact distance of the plane from the origin.