Analysis and Approaches for IBDP Maths Ebook 2 | Page 107

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Paper 2 Section B – Intersecting Lines and Planes
Example
A plane � has vector equation r � ( �i � j� 4 k) � �(2i �3 j�k) � �( i � 2 j�k ) , � , � � .
(a) Find the Cartesian equation of the plane � .
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(b) The plane � meets the x , y and z axes at A( a , 0, 0) , B(0, b , 0) and C(0, 0, c )
(c)
(d)
respectively. Write down the values of a , b and c .
Find the volume of the pyramid OABC , where O is the origin.
Find the vector equation of the line L that passes the point O and is
perpendicular to the plane ABC , giving the answer in parametric form.
(e) Find the exact perpendicular distance from the plane ABC to O .
(f) Hence, find the value of A , the area of the triangle ABC .
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Solution
(a) n � (2i �3 j�k) �( i � 2 j�k )
(M1) for valid approach
�(3)(1) � (1)(2) �
�
�
n �
(1)(1) �(2)(1)
�(2)(2) � (3)(1) �
�
�
n � i � j�k (A1) for correct values
The Cartesian equation of the plane � :
( xi � yj � zk) �( i � j�k) � ( �i � j� 4 k) �( i � j�k ) M1A1
x � y � z � ( �1)(1) � (1)( �1) � (4)(1)
x � y � z � 2
A1
(b) a � 2 , b �� 2 , c � 2
A3
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