Why vector product (CROSS PRODUCT) does not obey the commutative law like the scalar product (DOT PRODUCT)?

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Solution

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$T h e c r o s s p r o d u c t o f t w o v e c t o r s i s d e f i n e d a s (\vec{a} \times \vec{b}) = a b \sin θ \hat{n}, W h e r e t h e d i r e c t i o n o f C r o s s p r o d u c t i s g i v e n b y t h e r i g h t h a n d r u l e o f c r o s s p r o d u c t . A c c o r d i n g t o w h i c h s t r e t c h t h e f o r e f i n g e r o f t h e r i g h t h a n d i n t h e d i r e c t i o n o f \vec{a} a n d t h e m i d d l e f i n g e r i n t h e d i r e c t i o n o f \vec{b} . T h e n, t h e v e c t o r \hat{n} i s c o m i n g o u t o f t h e t h u m b w i l l r e p r e s e n t t h e d i r e c t i o n . B u t i f w e c h a n g e t h e v e c t o r s \vec{a} a n d \vec{b} t h e d i r e c t i o n o f t h e t h u m b w i l l b e o p p o s i t e t o t h e d i r e c t i o n a s i n t h e f i r s t c a s e . s o v e c t o r p r o d u c t d o n o t c o m m u t e .$

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