Notes On Addition of Vectors - CBSE Class 12 Maths
Triangle Law of Vector Addition: The sum of two vectors representing two sides of a triangle taken in the same order is given by the vector representing the third side of the triangle taken in the opposite order. Ex: Let vectors A and B be represented by the vectors AB and BC, respectively. By joining A and C, we get triangle ABC.    $\stackrel{\to }{\text{AB}}$   +     =  $\stackrel{\to }{\text{AC}}$       Reverse the direction of vector AC to get vector CA.  $\stackrel{\to }{\text{AB}}$   +     +     =  $\stackrel{\to }{\text{AA}}$    $\stackrel{\to }{\text{AB}}$   +     +     =  $\stackrel{\to }{\text{AA}}$   =     Thus, the sum of the vectors representing the three sides of a triangle in order is a zero vector. Here     = -     =     $\stackrel{\to }{\text{AC'}}$ = $\stackrel{\to }{\text{AB'}}$ + $\stackrel{\to }{\text{BC'}}$ ⇒ $\stackrel{\to }{\text{AC'}}$ = $\stackrel{\to }{\text{AB}}$ + (-$\stackrel{\to }{\text{BC}}$) ⇒ $\stackrel{\to }{\text{AC'}}$ = $\stackrel{\to }{\text{a}}$ +  (- $\stackrel{\to }{\text{b}}$) = $\stackrel{\to }{\text{a}}$ -  $\stackrel{\to }{\text{b}}$ Parallelogram Law of Vector Addition: The sum of two vectors representing adjacent sides of a parallelogram is given by the diagonal of the parallelogram passing through the common point of the two adjacent sides. $\stackrel{\to }{\text{OA}}$ + $\stackrel{\to }{\text{OB}}$ = $\stackrel{\to }{\text{OC}}$ ....parallelogram law $\stackrel{\to }{\text{AC}}$ = $\stackrel{\to }{\text{OB}}$ = $\stackrel{\to }{\text{b}}$ $\stackrel{\to }{\text{OA}}$ + $\stackrel{\to }{\text{AC}}$ = $\stackrel{\to }{\text{OC}}$ ....traingle law Thus, we see that the triangle and parallelogram laws of vector addition are equivalent to each other. Commutative Property of Vector Addition Given two vectors  $\stackrel{\to }{\text{a}}$   and  $\stackrel{\to }{\text{b}}$      : Let   $\stackrel{\to }{\text{AB}}$    = $\stackrel{\to }{\text{a}}$ and  $\stackrel{\to }{\text{BC}}$  = $\stackrel{\to }{\text{b}}$ By triangle law of vector addition: $\stackrel{\to }{\text{AB}}$ + $\stackrel{\to }{\text{BC}}$ = $\stackrel{\to }{\text{AC}}$ = $\stackrel{\to }{\text{a}}$   + $\stackrel{\to }{\text{b}}$...(1) In parallelogram ABCD: $\stackrel{\to }{\text{AD}}$ = $\stackrel{\to }{\text{BC}}$ = $\stackrel{\to }{\text{b}}$ $\stackrel{\to }{\text{DC}}$ = $\stackrel{\to }{\text{AB}}$ = $\stackrel{\to }{\text{a}}$ By triangle law of vector addition: $\stackrel{\to }{\text{AD}}$ + $\stackrel{\to }{\text{DC}}$ = $\stackrel{\to }{\text{AC}}$ = $\stackrel{\to }{\text{b}}$    + $\stackrel{\to }{\text{a}}$...(2) From (1) and (2):  $\stackrel{\to }{\text{a}}$  +  $\stackrel{\to }{\text{b}}$     = $\stackrel{\to }{\text{b}}$ + $\stackrel{\to }{\text{a}}$        Additive Identity: $\stackrel{\to }{\text{AB}}$   + $\stackrel{\to }{\text{0}}$   = $\stackrel{\to }{\text{0}}$  + $\stackrel{\to }{\text{AB}}$   = $\stackrel{\to }{\text{AB}}$ Associative Property of Vector Addition Given three vectors  $\stackrel{\to }{\text{a}}$ , $\stackrel{\to }{\text{b}}$    and $\stackrel{\to }{\text{c}}$     : (  $\stackrel{\to }{\text{a}}$   +  $\stackrel{\to }{\text{b}}$     ) + $\stackrel{\to }{\text{c}}$ = $\stackrel{\to }{\text{a}}$ + ( $\stackrel{\to }{\text{b}}$    +  $\stackrel{\to }{\text{c}}$ ) Consider the vectors   $\stackrel{\to }{\text{AB}}$    =  $\stackrel{\to }{\text{a}}$    ,  $\stackrel{\to }{\text{BC}}$   =   $\stackrel{\to }{\text{b}}$   and   $\stackrel{\to }{\text{CD}}$   =  $\stackrel{\to }{\text{c}}$     In ∆ ABC: $\stackrel{\to }{\text{AB}}$ + $\stackrel{\to }{\text{BC}}$ = $\stackrel{\to }{\text{AC}}$ =  $\stackrel{\to }{\text{a}}$   +  $\stackrel{\to }{\text{b}}$    [Triangle law of vector addition] In ∆ ACD: $\stackrel{\to }{\text{AC}}$ + $\stackrel{\to }{\text{CD}}$ = $\stackrel{\to }{\text{AD}}$ = ( $\stackrel{\to }{\text{a}}$    +   $\stackrel{\to }{\text{b}}$ ) + $\stackrel{\to }{\text{c}}$ ...(1) [Triangle law of vector addition] In ∆ BCD: $\stackrel{\to }{\text{BC}}$ + $\stackrel{\to }{\text{CD}}$ = $\stackrel{\to }{\text{BD}}$ = $\stackrel{\to }{\text{b}}$   + $\stackrel{\to }{\text{c}}$       [Triangle law of vector addition] In ∆ ABD: $\stackrel{\to }{\text{AB}}$ + $\stackrel{\to }{\text{BD}}$ = $\stackrel{\to }{\text{AD}}$ = $\stackrel{\to }{\text{a}}$ + ( $\stackrel{\to }{\text{b}}$   + $\stackrel{\to }{\text{c}}$ ) ...(2) [Triangle law of vector addition] From (1) and (2): ( $\stackrel{\to }{\text{a}}$   + $\stackrel{\to }{\text{b}}$ ) +  $\stackrel{\to }{\text{c}}$ =  $\stackrel{\to }{\text{a}}$ + ( $\stackrel{\to }{\text{b}}$   + $\stackrel{\to }{\text{c}}$ )

#### Summary

Triangle Law of Vector Addition: The sum of two vectors representing two sides of a triangle taken in the same order is given by the vector representing the third side of the triangle taken in the opposite order. Ex: Let vectors A and B be represented by the vectors AB and BC, respectively. By joining A and C, we get triangle ABC.    $\stackrel{\to }{\text{AB}}$   +     =  $\stackrel{\to }{\text{AC}}$       Reverse the direction of vector AC to get vector CA.  $\stackrel{\to }{\text{AB}}$   +     +     =  $\stackrel{\to }{\text{AA}}$    $\stackrel{\to }{\text{AB}}$   +     +     =  $\stackrel{\to }{\text{AA}}$   =     Thus, the sum of the vectors representing the three sides of a triangle in order is a zero vector. Here     = -     =     $\stackrel{\to }{\text{AC'}}$ = $\stackrel{\to }{\text{AB'}}$ + $\stackrel{\to }{\text{BC'}}$ ⇒ $\stackrel{\to }{\text{AC'}}$ = $\stackrel{\to }{\text{AB}}$ + (-$\stackrel{\to }{\text{BC}}$) ⇒ $\stackrel{\to }{\text{AC'}}$ = $\stackrel{\to }{\text{a}}$ +  (- $\stackrel{\to }{\text{b}}$) = $\stackrel{\to }{\text{a}}$ -  $\stackrel{\to }{\text{b}}$ Parallelogram Law of Vector Addition: The sum of two vectors representing adjacent sides of a parallelogram is given by the diagonal of the parallelogram passing through the common point of the two adjacent sides. $\stackrel{\to }{\text{OA}}$ + $\stackrel{\to }{\text{OB}}$ = $\stackrel{\to }{\text{OC}}$ ....parallelogram law $\stackrel{\to }{\text{AC}}$ = $\stackrel{\to }{\text{OB}}$ = $\stackrel{\to }{\text{b}}$ $\stackrel{\to }{\text{OA}}$ + $\stackrel{\to }{\text{AC}}$ = $\stackrel{\to }{\text{OC}}$ ....traingle law Thus, we see that the triangle and parallelogram laws of vector addition are equivalent to each other. Commutative Property of Vector Addition Given two vectors  $\stackrel{\to }{\text{a}}$   and  $\stackrel{\to }{\text{b}}$      : Let   $\stackrel{\to }{\text{AB}}$    = $\stackrel{\to }{\text{a}}$ and  $\stackrel{\to }{\text{BC}}$  = $\stackrel{\to }{\text{b}}$ By triangle law of vector addition: $\stackrel{\to }{\text{AB}}$ + $\stackrel{\to }{\text{BC}}$ = $\stackrel{\to }{\text{AC}}$ = $\stackrel{\to }{\text{a}}$   + $\stackrel{\to }{\text{b}}$...(1) In parallelogram ABCD: $\stackrel{\to }{\text{AD}}$ = $\stackrel{\to }{\text{BC}}$ = $\stackrel{\to }{\text{b}}$ $\stackrel{\to }{\text{DC}}$ = $\stackrel{\to }{\text{AB}}$ = $\stackrel{\to }{\text{a}}$ By triangle law of vector addition: $\stackrel{\to }{\text{AD}}$ + $\stackrel{\to }{\text{DC}}$ = $\stackrel{\to }{\text{AC}}$ = $\stackrel{\to }{\text{b}}$    + $\stackrel{\to }{\text{a}}$...(2) From (1) and (2):  $\stackrel{\to }{\text{a}}$  +  $\stackrel{\to }{\text{b}}$     = $\stackrel{\to }{\text{b}}$ + $\stackrel{\to }{\text{a}}$        Additive Identity: $\stackrel{\to }{\text{AB}}$   + $\stackrel{\to }{\text{0}}$   = $\stackrel{\to }{\text{0}}$  + $\stackrel{\to }{\text{AB}}$   = $\stackrel{\to }{\text{AB}}$ Associative Property of Vector Addition Given three vectors  $\stackrel{\to }{\text{a}}$ , $\stackrel{\to }{\text{b}}$    and $\stackrel{\to }{\text{c}}$     : (  $\stackrel{\to }{\text{a}}$   +  $\stackrel{\to }{\text{b}}$     ) + $\stackrel{\to }{\text{c}}$ = $\stackrel{\to }{\text{a}}$ + ( $\stackrel{\to }{\text{b}}$    +  $\stackrel{\to }{\text{c}}$ ) Consider the vectors   $\stackrel{\to }{\text{AB}}$    =  $\stackrel{\to }{\text{a}}$    ,  $\stackrel{\to }{\text{BC}}$   =   $\stackrel{\to }{\text{b}}$   and   $\stackrel{\to }{\text{CD}}$   =  $\stackrel{\to }{\text{c}}$     In ∆ ABC: $\stackrel{\to }{\text{AB}}$ + $\stackrel{\to }{\text{BC}}$ = $\stackrel{\to }{\text{AC}}$ =  $\stackrel{\to }{\text{a}}$   +  $\stackrel{\to }{\text{b}}$    [Triangle law of vector addition] In ∆ ACD: $\stackrel{\to }{\text{AC}}$ + $\stackrel{\to }{\text{CD}}$ = $\stackrel{\to }{\text{AD}}$ = ( $\stackrel{\to }{\text{a}}$    +   $\stackrel{\to }{\text{b}}$ ) + $\stackrel{\to }{\text{c}}$ ...(1) [Triangle law of vector addition] In ∆ BCD: $\stackrel{\to }{\text{BC}}$ + $\stackrel{\to }{\text{CD}}$ = $\stackrel{\to }{\text{BD}}$ = $\stackrel{\to }{\text{b}}$   + $\stackrel{\to }{\text{c}}$       [Triangle law of vector addition] In ∆ ABD: $\stackrel{\to }{\text{AB}}$ + $\stackrel{\to }{\text{BD}}$ = $\stackrel{\to }{\text{AD}}$ = $\stackrel{\to }{\text{a}}$ + ( $\stackrel{\to }{\text{b}}$   + $\stackrel{\to }{\text{c}}$ ) ...(2) [Triangle law of vector addition] From (1) and (2): ( $\stackrel{\to }{\text{a}}$   + $\stackrel{\to }{\text{b}}$ ) +  $\stackrel{\to }{\text{c}}$ =  $\stackrel{\to }{\text{a}}$ + ( $\stackrel{\to }{\text{b}}$   + $\stackrel{\to }{\text{c}}$ )

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