OCR MEI C4 (Core Mathematics 4)

1

1. Find the point of intersection of the line \(\left. \left. \mathbf { r } = \begin{array} { r } - 8
   - 2
   6 \end{array} \right) + \lambda \begin{array} { r } - 3
   0
   1 \end{array} \right)\) and the plane \(2 x - 3 y + z = 11\).
2. Find the acute angle between the line and the normal to the plane.

2 The points \(\mathrm { A } , \mathrm { B }\) and C have coordinates \(( 1,3 , - 2 ) , ( - 1,2 , - 3 )\) and \(( 0 , - 8,1 )\) respectively.

1. Find the vectors \(\overrightarrow { \mathrm { AB } }\) and \(\overrightarrow { \mathrm { AC } }\).
2. Show that the vector \(2 \mathbf { i } - \mathbf { j } - 3 \mathbf { k }\) is perpendicular to the plane ABC . Hence find the equation of the plane ABC .
3. Write down normal vectors to the planes \(2 x - y + z = 2\) and \(x - z = 1\). Hence find the acute angle between the planes.
4. Write down a vector equation of the line through \(( 2,0,1 )\) perpendicular to the plane \(2 x - y + z = 2\). Find the point of intersection of this line with the plane.
5. Find the cartesian equation of the plane through the point \(( 2 , - 1,4 )\) with normal vector $$\mathbf { n } = \left( \begin{array} { l } 1
   1
   2 \end{array} \right) .$$
6. Find the coordinates of the point of intersection of this plane and the straight line with equation $$\mathbf { r } = \left( \begin{array} { r } 7
   12
   9 \end{array} \right) + \lambda \left( \begin{array} { l } 1

12
9 \end{array} \right) + \lambda \left( \begin{array} { l } 1
3
2 \end{array} \right)$$ \begin{figure}[h] \end{figure} Fig. 7 illustrates a house. All units are in metres. The coordinates of A, B, C and E are as shown. BD is horizontal and parallel to AE .

1. Find the length AE .
2. Find a vector equation of the line BD . Given that the length of BD is 15 metres, find the coordinates of D.
3. Verify that the equation of the plane ABC is $$- 3 x + 4 y + 5 z = 30 .$$ Write down a vector normal to this plane.
4. Show that the vector \(\left( \begin{array} { l } 4
   3
   5 \end{array} \right)\) is normal to the plane ABDE . Hence find the equation of the plane ABDE .
5. Find the angle between the planes ABC and ABDE .