Notes On Cross Product - CBSE Class 12 Maths
The understanding of the right handed coordinate system is essential to understand the vector or cross product of two vectors. Given $\stackrel{â†’}{\text{p}}$ â‰   $\stackrel{â†’}{\text{0}}$ and  $\stackrel{â†’}{\text{q}}$ â‰  $\stackrel{â†’}{\text{0}}$ Let Ï´ be the angle between $\stackrel{â†’}{\text{p}}$ and $\stackrel{â†’}{\text{q}}$ such that: 0o â‰¤ Ï´ â‰¤ 180o. Cross product of $\stackrel{â†’}{\text{p}}$ and $\stackrel{â†’}{\text{q}}$ is given by: $\stackrel{â†’}{\text{p}}$ x $\stackrel{â†’}{\text{q}}$ =  |$\stackrel{â†’}{\text{p}}$| |$\stackrel{â†’}{\text{q}}$|  sin Ï´ $\stackrel{^}{\text{n}}$ Where $\stackrel{^}{\text{n}}$  âŠ¥ $\stackrel{â†’}{\text{p}}$ and  $\stackrel{^}{\text{n}}$  âŠ¥ $\stackrel{â†’}{\text{q}}$ Since the cross product of two vectors is also a vector, it is also called a vector product. If $\stackrel{â†’}{\text{p}}$  = $\stackrel{â†’}{\text{0}}$    ,   â‡’ |$\stackrel{â†’}{\text{p}}$| = 0       Similarly, if   $\stackrel{â†’}{\text{q}}$  = $\stackrel{â†’}{\text{0}}$    â‡’ |$\stackrel{â†’}{\text{q}}$| = 0    Thus, if either $\stackrel{â†’}{\text{p}}$ or $\stackrel{â†’}{\text{q}}$ = $\stackrel{â†’}{\text{0}}$,  $\stackrel{â†’}{\text{p}}$  x  $\stackrel{â†’}{\text{q}}$   = |$\stackrel{â†’}{\text{p}}$| |$\stackrel{â†’}{\text{q}}$|  sin Ï´  $\stackrel{^}{\text{n}}$ = $\stackrel{â†’}{\text{0}}$

#### Summary

The understanding of the right handed coordinate system is essential to understand the vector or cross product of two vectors. Given $\stackrel{â†’}{\text{p}}$ â‰   $\stackrel{â†’}{\text{0}}$ and  $\stackrel{â†’}{\text{q}}$ â‰  $\stackrel{â†’}{\text{0}}$ Let Ï´ be the angle between $\stackrel{â†’}{\text{p}}$ and $\stackrel{â†’}{\text{q}}$ such that: 0o â‰¤ Ï´ â‰¤ 180o. Cross product of $\stackrel{â†’}{\text{p}}$ and $\stackrel{â†’}{\text{q}}$ is given by: $\stackrel{â†’}{\text{p}}$ x $\stackrel{â†’}{\text{q}}$ =  |$\stackrel{â†’}{\text{p}}$| |$\stackrel{â†’}{\text{q}}$|  sin Ï´ $\stackrel{^}{\text{n}}$ Where $\stackrel{^}{\text{n}}$  âŠ¥ $\stackrel{â†’}{\text{p}}$ and  $\stackrel{^}{\text{n}}$  âŠ¥ $\stackrel{â†’}{\text{q}}$ Since the cross product of two vectors is also a vector, it is also called a vector product. If $\stackrel{â†’}{\text{p}}$  = $\stackrel{â†’}{\text{0}}$    ,   â‡’ |$\stackrel{â†’}{\text{p}}$| = 0       Similarly, if   $\stackrel{â†’}{\text{q}}$  = $\stackrel{â†’}{\text{0}}$    â‡’ |$\stackrel{â†’}{\text{q}}$| = 0    Thus, if either $\stackrel{â†’}{\text{p}}$ or $\stackrel{â†’}{\text{q}}$ = $\stackrel{â†’}{\text{0}}$,  $\stackrel{â†’}{\text{p}}$  x  $\stackrel{â†’}{\text{q}}$   = |$\stackrel{â†’}{\text{p}}$| |$\stackrel{â†’}{\text{q}}$|  sin Ï´  $\stackrel{^}{\text{n}}$ = $\stackrel{â†’}{\text{0}}$

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