� Properties of the vector product u� v of angle between u and v :
�u1
� � � u � u2 and
�u
� � 3 �
1 . |
�u2v3
� u3v2
�
�
� u� v
� u
ˆ 3v1
� u1v3
� u v sin�n
, where nˆ //( u� v
)
�u1v2
� u2v
�
�
1
�
|
2 . |
i�i � j� j � k
�k
�
0
|
3 . i�j �k , j�k �i and k �i � j 4 . j�i � �k , k � j � �i and i�k
� �j
5 . u and v are parallel if u�v
�0 6 . u and v are perpendicular if u�v � u v 7 . u� v � �( v�u
)
�v1
� � � v � v2 where � is the
�v
� � 3 �
� The area of the parallelogram with adjacent sides AB � and AD � � � is AB�
AD
� The area of the triangle with adjacent sides AB � and AD � is 1 � � AB�
AD 2
� 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�
� Vector components : 1 .
2 .
� x �a1�b1t � �y � a2 � b2t
: Parametric form
� � z � a3 � b3t
u� v
: Vector component of u parallel to v v u� v v
: Vector component of u perpendicular to v
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