Laws of vector operations

Vector operations, like many other mathematical objects, obey a few laws:

1. Vector addition
   1. associativity
      • $(\vec{A}+\vec{B})+\vec{C}=\vec{A}+(\vec{B}+\vec{C})$
   2. commutativity
      • $\vec{A}+\vec{B}=\vec{B}+\vec{A}$
   3. identity element
      • $\vec{A} + \vec{0} = \vec{0} + \vec{A} = \vec{A}
   4. Inverse element
      • $\vec{A}+(-\vec{A})=0$
2. Scalar multiplication
   1. associativity
   2. distributive law
   3. Distributive law for scalar addition
      • $(b+c)\vec{A}=b\vec{A}+c\vec{A}$
   4. identity element
      • $1\vec{A}=\vec{A}$
3. Unit vector
   • Dividing a vector by its magnitude results in a vector of unit length, denoted by the caret symbol $\hat{A}=\frac{\vec{A}}{|\vec{A}|}$

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