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 $\stackrel{\to }{\text{p}}$ ≠  $\stackrel{\to }{\text{0}}$ and  $\stackrel{\to }{\text{q}}$ ≠ $\stackrel{\to }{\text{0}}$ Let ϴ be the angle between $\stackrel{\to }{\text{p}}$ and $\stackrel{\to }{\text{q}}$ such that: 0o ≤ ϴ ≤ 180o. Cross product of $\stackrel{\to }{\text{p}}$ and $\stackrel{\to }{\text{q}}$ is given by: $\stackrel{\to }{\text{p}}$ x $\stackrel{\to }{\text{q}}$ =  |$\stackrel{\to }{\text{p}}$| |$\stackrel{\to }{\text{q}}$|  sin ϴ $\stackrel{^}{\text{n}}$ Where $\stackrel{^}{\text{n}}$  ⊥ $\stackrel{\to }{\text{p}}$ and  $\stackrel{^}{\text{n}}$  ⊥ $\stackrel{\to }{\text{q}}$ Since the cross product of two vectors is also a vector, it is also called a vector product. If $\stackrel{\to }{\text{p}}$  = $\stackrel{\to }{\text{0}}$    ,   ⇒ |$\stackrel{\to }{\text{p}}$| = 0       Similarly, if   $\stackrel{\to }{\text{q}}$  = $\stackrel{\to }{\text{0}}$    ⇒ |$\stackrel{\to }{\text{q}}$| = 0    Thus, if either $\stackrel{\to }{\text{p}}$ or $\stackrel{\to }{\text{q}}$ = $\stackrel{\to }{\text{0}}$,  $\stackrel{\to }{\text{p}}$  x  $\stackrel{\to }{\text{q}}$   = |$\stackrel{\to }{\text{p}}$| |$\stackrel{\to }{\text{q}}$|  sin ϴ  $\stackrel{^}{\text{n}}$ = $\stackrel{\to }{\text{0}}$

#### Summary

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

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