Analysis and Approaches for IBDP Maths Ebook 2 | Page 109

Exercise 32
1. A plane � has vector equation r � ( �6i �18 k) � �(3i � 2 j) � �( j�3 k ) , � , � � .
(a) Find the Cartesian equation of the plane � .
� 1 � � �10�
� � � �
(b) Find the coordinates of the point of intersection where the line r �
3 �t
5
�13� � �13�
� � � �
meets the plane � .
[3]
(c) The plane � meets the x , y and z axes at A( a , 0, 0) , B(0, b , 0) and C(0, 0, c )
(d)
respectively. Write down the values of a , b and c .
Find the volume of the pyramid OABC , where O is the origin.
[5]
[3]
[3]
(e) (i) Write down CA � and CB
� .
(f)
(ii) Hence, find the value of � if the area of the triangle ABC is � 14 .
[6]
Use the results of (d) and (e), find the exact perpendicular distance from the plane
ABC to O .
[2]
8
2. A plane � has vector equation r � ( �6i � 2 j) � �(3 i � j) � �(4 i �k ) , � , � � .
(a) Find the Cartesian equation of the plane � .
[5]
(b) The plane � meets the x , y and z axes at A , B and C respectively. Let O be
(c)
(d)
the origin. Find the volume of the pyramid OABC .
Find the vector equation of the line L that passes the point A and is
perpendicular to the plane ABC .
Let A� be the reflection point of A about the y - z plane.
[6]
[2]
(i)
(ii)
Write down the coordinates of A�.
Find the vector equation of the line L� that passes the point A� and is
perpendicular to the plane ABC , giving the answer in Cartesian form.
(iii) The point ( � �72, � � 12, �)
lies on L� . Find the value of � .
www.seprodstore.com
[5]
99