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.    $\stackrel{â†’}{\text{AB}}$   +     =  $\stackrel{â†’}{\text{AC}}$       Reverse the direction of vector AC to get vector CA.  $\stackrel{â†’}{\text{AB}}$   +     +     =  $\stackrel{â†’}{\text{AA}}$    $\stackrel{â†’}{\text{AB}}$   +     +     =  $\stackrel{â†’}{\text{AA}}$   =     Thus, the sum of the vectors representing the three sides of a triangle in order is a zero vector. Here     = -     =     $\stackrel{â†’}{\text{AC'}}$ = $\stackrel{â†’}{\text{AB'}}$ + $\stackrel{â†’}{\text{BC'}}$ â‡’ $\stackrel{â†’}{\text{AC'}}$ = $\stackrel{â†’}{\text{AB}}$ + (-$\stackrel{â†’}{\text{BC}}$) â‡’ $\stackrel{â†’}{\text{AC'}}$ = $\stackrel{â†’}{\text{a}}$ +  (- $\stackrel{â†’}{\text{b}}$) = $\stackrel{â†’}{\text{a}}$ -  $\stackrel{â†’}{\text{b}}$ 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. $\stackrel{â†’}{\text{OA}}$ + $\stackrel{â†’}{\text{OB}}$ = $\stackrel{â†’}{\text{OC}}$ ....parallelogram law $\stackrel{â†’}{\text{AC}}$ = $\stackrel{â†’}{\text{OB}}$ = $\stackrel{â†’}{\text{b}}$ $\stackrel{â†’}{\text{OA}}$ + $\stackrel{â†’}{\text{AC}}$ = $\stackrel{â†’}{\text{OC}}$ ....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  $\stackrel{â†’}{\text{a}}$   and  $\stackrel{â†’}{\text{b}}$      : Let   $\stackrel{â†’}{\text{AB}}$    = $\stackrel{â†’}{\text{a}}$ and  $\stackrel{â†’}{\text{BC}}$  = $\stackrel{â†’}{\text{b}}$              By triangle law of vector addition: $\stackrel{â†’}{\text{AB}}$ + $\stackrel{â†’}{\text{BC}}$ = $\stackrel{â†’}{\text{AC}}$ = $\stackrel{â†’}{\text{a}}$   + $\stackrel{â†’}{\text{b}}$...(1) In parallelogram ABCD: $\stackrel{â†’}{\text{AD}}$ = $\stackrel{â†’}{\text{BC}}$ = $\stackrel{â†’}{\text{b}}$ $\stackrel{â†’}{\text{DC}}$ = $\stackrel{â†’}{\text{AB}}$ = $\stackrel{â†’}{\text{a}}$ By triangle law of vector addition: $\stackrel{â†’}{\text{AD}}$ + $\stackrel{â†’}{\text{DC}}$ = $\stackrel{â†’}{\text{AC}}$ = $\stackrel{â†’}{\text{b}}$    + $\stackrel{â†’}{\text{a}}$...(2) From (1) and (2):  $\stackrel{â†’}{\text{a}}$  +  $\stackrel{â†’}{\text{b}}$     = $\stackrel{â†’}{\text{b}}$ + $\stackrel{â†’}{\text{a}}$        Additive Identity: $\stackrel{â†’}{\text{AB}}$   + $\stackrel{â†’}{\text{0}}$   = $\stackrel{â†’}{\text{0}}$  + $\stackrel{â†’}{\text{AB}}$   = $\stackrel{â†’}{\text{AB}}$ Associative Property of Vector Addition Given three vectors  $\stackrel{â†’}{\text{a}}$ , $\stackrel{â†’}{\text{b}}$    and $\stackrel{â†’}{\text{c}}$     : (  $\stackrel{â†’}{\text{a}}$   +  $\stackrel{â†’}{\text{b}}$     ) + $\stackrel{â†’}{\text{c}}$ = $\stackrel{â†’}{\text{a}}$ + ( $\stackrel{â†’}{\text{b}}$    +  $\stackrel{â†’}{\text{c}}$ ) Consider the vectors   $\stackrel{â†’}{\text{AB}}$    =  $\stackrel{â†’}{\text{a}}$    ,  $\stackrel{â†’}{\text{BC}}$   =   $\stackrel{â†’}{\text{b}}$   and   $\stackrel{â†’}{\text{CD}}$   =  $\stackrel{â†’}{\text{c}}$     In âˆ† ABC: $\stackrel{â†’}{\text{AB}}$ + $\stackrel{â†’}{\text{BC}}$ = $\stackrel{â†’}{\text{AC}}$ =  $\stackrel{â†’}{\text{a}}$   +  $\stackrel{â†’}{\text{b}}$    [Triangle law of vector addition] In âˆ† ACD: $\stackrel{â†’}{\text{AC}}$ + $\stackrel{â†’}{\text{CD}}$ = $\stackrel{â†’}{\text{AD}}$ = ( $\stackrel{â†’}{\text{a}}$    +   $\stackrel{â†’}{\text{b}}$ ) + $\stackrel{â†’}{\text{c}}$ ...(1) [Triangle law of vector addition] In âˆ† BCD: $\stackrel{â†’}{\text{BC}}$ + $\stackrel{â†’}{\text{CD}}$ = $\stackrel{â†’}{\text{BD}}$ = $\stackrel{â†’}{\text{b}}$   + $\stackrel{â†’}{\text{c}}$       [Triangle law of vector addition] In âˆ† ABD: $\stackrel{â†’}{\text{AB}}$ + $\stackrel{â†’}{\text{BD}}$ = $\stackrel{â†’}{\text{AD}}$ = $\stackrel{â†’}{\text{a}}$ + ( $\stackrel{â†’}{\text{b}}$   + $\stackrel{â†’}{\text{c}}$ ) ...(2) [Triangle law of vector addition] From (1) and (2): ( $\stackrel{â†’}{\text{a}}$   + $\stackrel{â†’}{\text{b}}$ ) +  $\stackrel{â†’}{\text{c}}$ =  $\stackrel{â†’}{\text{a}}$ + ( $\stackrel{â†’}{\text{b}}$   + $\stackrel{â†’}{\text{c}}$ )

#### 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.    $\stackrel{â†’}{\text{AB}}$   +     =  $\stackrel{â†’}{\text{AC}}$       Reverse the direction of vector AC to get vector CA.  $\stackrel{â†’}{\text{AB}}$   +     +     =  $\stackrel{â†’}{\text{AA}}$    $\stackrel{â†’}{\text{AB}}$   +     +     =  $\stackrel{â†’}{\text{AA}}$   =     Thus, the sum of the vectors representing the three sides of a triangle in order is a zero vector. Here     = -     =     $\stackrel{â†’}{\text{AC'}}$ = $\stackrel{â†’}{\text{AB'}}$ + $\stackrel{â†’}{\text{BC'}}$ â‡’ $\stackrel{â†’}{\text{AC'}}$ = $\stackrel{â†’}{\text{AB}}$ + (-$\stackrel{â†’}{\text{BC}}$) â‡’ $\stackrel{â†’}{\text{AC'}}$ = $\stackrel{â†’}{\text{a}}$ +  (- $\stackrel{â†’}{\text{b}}$) = $\stackrel{â†’}{\text{a}}$ -  $\stackrel{â†’}{\text{b}}$ 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. $\stackrel{â†’}{\text{OA}}$ + $\stackrel{â†’}{\text{OB}}$ = $\stackrel{â†’}{\text{OC}}$ ....parallelogram law $\stackrel{â†’}{\text{AC}}$ = $\stackrel{â†’}{\text{OB}}$ = $\stackrel{â†’}{\text{b}}$ $\stackrel{â†’}{\text{OA}}$ + $\stackrel{â†’}{\text{AC}}$ = $\stackrel{â†’}{\text{OC}}$ ....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  $\stackrel{â†’}{\text{a}}$   and  $\stackrel{â†’}{\text{b}}$      : Let   $\stackrel{â†’}{\text{AB}}$    = $\stackrel{â†’}{\text{a}}$ and  $\stackrel{â†’}{\text{BC}}$  = $\stackrel{â†’}{\text{b}}$              By triangle law of vector addition: $\stackrel{â†’}{\text{AB}}$ + $\stackrel{â†’}{\text{BC}}$ = $\stackrel{â†’}{\text{AC}}$ = $\stackrel{â†’}{\text{a}}$   + $\stackrel{â†’}{\text{b}}$...(1) In parallelogram ABCD: $\stackrel{â†’}{\text{AD}}$ = $\stackrel{â†’}{\text{BC}}$ = $\stackrel{â†’}{\text{b}}$ $\stackrel{â†’}{\text{DC}}$ = $\stackrel{â†’}{\text{AB}}$ = $\stackrel{â†’}{\text{a}}$ By triangle law of vector addition: $\stackrel{â†’}{\text{AD}}$ + $\stackrel{â†’}{\text{DC}}$ = $\stackrel{â†’}{\text{AC}}$ = $\stackrel{â†’}{\text{b}}$    + $\stackrel{â†’}{\text{a}}$...(2) From (1) and (2):  $\stackrel{â†’}{\text{a}}$  +  $\stackrel{â†’}{\text{b}}$     = $\stackrel{â†’}{\text{b}}$ + $\stackrel{â†’}{\text{a}}$        Additive Identity: $\stackrel{â†’}{\text{AB}}$   + $\stackrel{â†’}{\text{0}}$   = $\stackrel{â†’}{\text{0}}$  + $\stackrel{â†’}{\text{AB}}$   = $\stackrel{â†’}{\text{AB}}$ Associative Property of Vector Addition Given three vectors  $\stackrel{â†’}{\text{a}}$ , $\stackrel{â†’}{\text{b}}$    and $\stackrel{â†’}{\text{c}}$     : (  $\stackrel{â†’}{\text{a}}$   +  $\stackrel{â†’}{\text{b}}$     ) + $\stackrel{â†’}{\text{c}}$ = $\stackrel{â†’}{\text{a}}$ + ( $\stackrel{â†’}{\text{b}}$    +  $\stackrel{â†’}{\text{c}}$ ) Consider the vectors   $\stackrel{â†’}{\text{AB}}$    =  $\stackrel{â†’}{\text{a}}$    ,  $\stackrel{â†’}{\text{BC}}$   =   $\stackrel{â†’}{\text{b}}$   and   $\stackrel{â†’}{\text{CD}}$   =  $\stackrel{â†’}{\text{c}}$     In âˆ† ABC: $\stackrel{â†’}{\text{AB}}$ + $\stackrel{â†’}{\text{BC}}$ = $\stackrel{â†’}{\text{AC}}$ =  $\stackrel{â†’}{\text{a}}$   +  $\stackrel{â†’}{\text{b}}$    [Triangle law of vector addition] In âˆ† ACD: $\stackrel{â†’}{\text{AC}}$ + $\stackrel{â†’}{\text{CD}}$ = $\stackrel{â†’}{\text{AD}}$ = ( $\stackrel{â†’}{\text{a}}$    +   $\stackrel{â†’}{\text{b}}$ ) + $\stackrel{â†’}{\text{c}}$ ...(1) [Triangle law of vector addition] In âˆ† BCD: $\stackrel{â†’}{\text{BC}}$ + $\stackrel{â†’}{\text{CD}}$ = $\stackrel{â†’}{\text{BD}}$ = $\stackrel{â†’}{\text{b}}$   + $\stackrel{â†’}{\text{c}}$       [Triangle law of vector addition] In âˆ† ABD: $\stackrel{â†’}{\text{AB}}$ + $\stackrel{â†’}{\text{BD}}$ = $\stackrel{â†’}{\text{AD}}$ = $\stackrel{â†’}{\text{a}}$ + ( $\stackrel{â†’}{\text{b}}$   + $\stackrel{â†’}{\text{c}}$ ) ...(2) [Triangle law of vector addition] From (1) and (2): ( $\stackrel{â†’}{\text{a}}$   + $\stackrel{â†’}{\text{b}}$ ) +  $\stackrel{â†’}{\text{c}}$ =  $\stackrel{â†’}{\text{a}}$ + ( $\stackrel{â†’}{\text{b}}$   + $\stackrel{â†’}{\text{c}}$ )

Previous
Next
âž¤