Equation of plane through three points

9
The diagram shows three points \(A , B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 1 \\ 2 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 0 \\ 3 \\ 1 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 3 \\ 0 \\ 4 \end{array} \right)\). The point \(D\) lies on \(B C\), between \(B\) and \(C\), and is such that \(C D = 2 D B\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the position vector of \(D\).
3. Show that the length of the perpendicular from \(A\) to \(O D\) is \(\frac { 1 } { 3 } \sqrt { } ( 65 )\).

2 The points \(A , B , C\) have position vectors $$4 \mathbf { i } - 4 \mathbf { j } + \mathbf { k } , \quad - 4 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } , \quad 4 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ,$$ respectively, relative to the origin \(O\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the perpendicular distance from \(O\) to the plane \(A B C\).
3. The point \(D\) has position vector \(2 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k }\). Find the coordinates of the point of intersection of the line \(O D\) with the plane \(A B C\).

6 The points \(A , B , C\) have position vectors $$\mathbf { i } + \mathbf { j } , \quad - \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , \quad - 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ,$$ respectively, relative to the origin \(O\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the perpendicular distance from \(O\) to the plane \(A B C\).
3. Find a vector equation of the common perpendicular to the lines \(O C\) and \(A B\).

5 The points \(A , B , C\) have position vectors $$2 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , \quad 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad - 3 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$$ respectively, relative to the origin \(O\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
   The point \(D\) has position vector \(2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }\).
2. Find the perpendicular distance from \(D\) to the plane \(A B C\).
3. Find the shortest distance between the lines \(A B\) and \(C D\).

4 The points \(A , B , C\) have position vectors $$- \mathbf { i } + \mathbf { j } + 2 \mathbf { k } , \quad - 2 \mathbf { i } - \mathbf { j } , \quad 2 \mathbf { i } + 2 \mathbf { k } ,$$ respectively, relative to the origin \(O\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the perpendicular distance from \(O\) to the plane \(A B C\).
3. Find the acute angle between the planes \(O A B\) and \(A B C\).

3. The position vectors of the points \(A , B\) and \(C\) relative to an origin \(O\) are \(\mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } , 7 \mathbf { i } - 3 \mathbf { k }\) and \(4 \mathbf { i } + 4 \mathbf { j }\) respectively. Find

1. \(\overrightarrow { A C } \times \overrightarrow { B C }\),
2. the area of triangle \(A B C\),
3. an equation of the plane \(A B C\) in the form \(\mathbf { r } . \mathbf { n } = p\)

1 A mine contains several underground tunnels beneath a hillside. The hillside is a plane, all the tunnels are straight and the width of the tunnels may be neglected. A coordinate system is chosen with the \(z\)-axis pointing vertically upwards and the units are metres. Three points on the hillside have coordinates \(\mathrm { A } ( 15 , - 60,20 )\), \(B ( - 75,100,40 )\) and \(C ( 18,138,35.6 )\).

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\) and hence show that the equation of the hillside is \(2 x - 2 y + 25 z = 650\). The tunnel \(T _ { \mathrm { A } }\) begins at A and goes in the direction of the vector \(15 \mathbf { i } + 14 \mathbf { j } - 2 \mathbf { k }\); the tunnel \(T _ { \mathrm { C } }\) begins at C and goes in the direction of the vector \(8 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\). Both these tunnels extend a long way into the ground.
2. Find the least possible length of a tunnel which connects B to a point in \(T _ { \mathrm { A } }\).
3. Find the least possible length of a tunnel which connects a point in \(T _ { \mathrm { A } }\) to a point in \(T _ { \mathrm { C } }\).
4. A tunnel starts at B , passes through the point ( \(18,138 , p\) ) vertically below C , and intersects \(T _ { \mathrm { A } }\) at the point Q . Find the value of \(p\) and the coordinates of Q .

1 Points \(A , B\) and \(C\) have coordinates \(( 0,1 , - 4 ) , ( 1,1 , - 2 )\) and \(( 3,2,5 )\) respectively.

1. Find the vector product \(\overrightarrow { A B } \times \overrightarrow { A C }\).
2. Hence find the equation of the plane \(A B C\) in the form \(a x + b y + c z = d\).

9 Three non-collinear points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to the origin \(O\). The plane through \(A , B\) and \(C\) is denoted by \(\Pi\).

1. (a) Prove that the area of triangle \(A B C\) is \(\frac { 1 } { 2 } | \mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a } |\).
   (b) Describe the significance of the vector \(\mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a }\) in relation to \(\Pi\).
2. (a) In the case when \(\mathbf { a } = a \mathbf { i } , \mathbf { b } = b \mathbf { j }\) and \(\mathbf { c } = c \mathbf { k }\), where \(a , b\) and \(c\) are positive scalar constants, determine the equation of \(\Pi\) in the form r.n \(= d\), where the components of \(\mathbf { n }\) and the value of the scalar constant \(d\) are to be given in terms of \(a , b\) and \(c\).
   (b) Deduce the shortest distance from the origin \(O\) to \(\Pi\) in this case.

\includegraphics{figure_9} The diagram shows three points \(A\), \(B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow{OA} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}\), \(\overrightarrow{OB} = \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}\) and \(\overrightarrow{OC} = \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}\). The point \(D\) lies on \(BC\), between \(B\) and \(C\), and is such that \(CD = 2DB\).

1. Find the equation of the plane \(ABC\), giving your answer in the form \(ax + by + cz = d\). [6]
2. Find the position vector of \(D\). [1]
3. Show that the length of the perpendicular from \(A\) to \(OD\) is \(\frac{1}{3}\sqrt{(65)}\). [4]

Points \(A\), \(B\) and \(C\) have coordinates \((0, 1, -4)\), \((1, 1, -2)\) and \((3, 2, 5)\) respectively.

1. Find the vector product \(\overrightarrow{AB} \times \overrightarrow{AC}\). [3]
2. Hence find the equation of the plane \(ABC\) in the form \(ax + by + cz = d\). [2]