Analysis and Approaches for IBDP Maths Ebook 2 | Page 105

31
Paper 2 Section A – Intersection of Planes
Example
Find the acute angle between the planes with equations x �3y �5z
� 7 and
2x � 4y �3z
� 0.
[5]
Solution
Let n
1
and n
2
be the normal vectors of x �3y �5z
� 7 and
2x � 4y �3z
� 0 respectively.
n
1
�1�
� �
� 3
�5�
� �
� 2 �
� �
n
2
� �4
(A1) for correct values
� 3 �
� �
Let � be the acute angle between the planes.
n � n � n n (M1) for valid approach
1 2 1 2 cos�
(1)(2) � (3)( �4) � (5)(3)
� � � � � �
2 2 2 2 2 2
( 1 3 5 )( 2 ( 4) 3 )cos�
5 � ( 35)( 29)cos�
5
cos� � A1
( 35)( 29)
� � 80.97060282
� � 81.0
A1
(A1) for substitution
[5]
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