Analysis and Approaches HL Practice Paper Book | Page 32

Your Practice Paper – Analysis and Approaches HL for IBDP Mathematics

� Forms of the straight line with fixed point A ( a1, a2, a

3) and direction vector b � b i � b j� b k :

1 .

2 .

1 2 3

� x � � a1� � b1� � � � � � � y � a �t b , t � 2 2

� z � � a � �

3 b � � � � � � 3� � x �a1�b1t � �y _� a2 _� b2t

: Parametric form

� � z _� a3 _� b3t x �a1 y �a2 z� a3 3 . � � ( � t)

: Cartesian equations b b b

1 2 3

� Intersections of two lines : 1 . Intersect at one point ( One intersection ) 2 . Skew ( No intersection ) 3 . Parallel ( No intersection ) 4 . Coincide ( Infinite number of intersections )

� Forms of the plane with fixed point A ( a1, a2, a

3) and normal vector

1 .

2 .

� x � � n1 � � a1 � � n1

� � � � � � � � � y � n � a � n

2 2 2

� z � � n � �

3 a � �

3 n � � � � � � � � 3 �

2

� z� � a � � � � 3 �

�n1

� � � n � n2 :

�n

� � 3 �

� x� � a1

� � � � � y � a � �u�

�v , � , � � , where u and v are two non-parallel

vectors on the plane

3 . n1 x � n2 y � n3z � a1n 1 � a2n2 � a3n3

: Cartesian form

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