Vector Operations>Triangle Law for Vector Addition

Vectors are physical quantities that have both magnitude and direction. When two vectors don't act along the same line, we can't just add them like regular numbers. The triangle law of vector addition is a method for finding the sum, also known as the resultant. This law states that if two vectors are represented as two sides of a triangle taken in sequence, their resultant is the third side of the triangle taken in the opposite direction. This resultant vector closes the triangle, running from the start of the first vector to the end of the second.

Two \[\vec A\] vectors \[\vec B\] and in a plane

Resultant vector \[\vec C\] = \[\vec A\] + \[\vec B\]

This law states that the order in which you add two vectors does not change the final result. For any two vectors \[\vec P\] and \[\vec Q\], the formula is:
\[\vec P\] + \[\vec Q\] = \[\vec Q\] + \[\vec P\] 
Step 1: To find \[\vec{P}\] + \[\vec{Q}\] , we can form a triangle OCB. Here,  
\[\overrightarrow {CB}\] = \[\vec P\] and  \[\overrightarrow {OC}\] = \[\vec Q\]. The resultant is \[\vec R\] = \[\overrightarrow {OB}\] .
Step 2: To find \[\vec{Q}\] + \[\vec{P}\], we can form a triangle OAB. Here,  \[\overrightarrow {OA}\] = \[\vec P\]  and  \[\overrightarrow {AB}\] = \[\vec Q\] . The resultant is also \[\vec R\] = \[\overrightarrow {OB}\].
Conclusion: Since both methods give the same resultant vector (\[\overrightarrow {OB}\]), we can conclude that vector addition is commutative.

Commutative law

This law states that, when adding three or more vectors, how you group them does not change the final result. For any three vectors \[\vec{A}\], \[\vec{B}\], and \[\vec{C}\], the formula is:
(\[\vec A\] + \[\vec B\]) + \[\vec C\] = \[\vec A\] + (\[\vec B\] + \[\vec C\])

Step 1: Grouping \[\vec A\] and \[\vec B\] first. First, add \[\vec A\] and \[\vec B\] to get a resultant, shown as \[\overrightarrow {OQ}\] in the diagram.
Then, add vector \[\vec C\] to this resultant (\[\overrightarrow {OQ}\]) to get the final resultant \[\vec R\].
This represents ( A + B)+ C.

Step 2: Grouping \[\vec B\] and \[\vec C\] first. First, add \[\vec B\] and \[\vec C\] to get a resultant, shown as \[\overrightarrow {PR}\] in the diagram. Then, add this resultant (\[\overrightarrow {PR}\]) to vector \[\vec A\] to get the final resultant \[\vec R\].
This represents \[\vec A\] + (\[\vec B\] + \[\vec C\]).

Conclusion: Both grouping methods result in the same final vector \[\vec{R}\], proving that vector addition is associative.

Associative law

Question: Express vector \[\overrightarrow {AC}\] in terms of vectors \[\overrightarrow {AB}\] and \[\overrightarrow {CB}\] shown in the following figure.

Solution:

According to the triangle law, if we follow the vectors in order, \[\overrightarrow {AC}\] followed by \[\overrightarrow {CB}\] gives the resultant vector \[\overrightarrow {AB}\].

We can write this as an equation: \[\overrightarrow {AC}\] + \[\overrightarrow {CB}\] = \[\overrightarrow {AB}\]

To find \[\overrightarrow{AC}\], we rearrange the equation:
\[\overrightarrow {AC}\] = \[\overrightarrow {AB}\] − \[\overrightarrow {CB}\]

Question: From the following figure, determine the resultant of four forces \[\vec{A_1}\], \[\vec{A_2}\], \[\vec{A_3}\], and \[\vec{A_4}\].

Solution: We can find the resultant by adding the vectors one by one using the triangle law.

Step 1: Add  \[\vec {A_1}\] ​and \[\vec {A_2}\].
\[\overrightarrow {OB}\] = \[\overrightarrow {OA}\] + \[\overrightarrow {AB}\]
\[\overrightarrow {OB}\] = \[\vec {A_1}\] + \[\vec {A_2}\]

Fig.(a)

Step 2: Add \[\vec {A_3}\] to the result from Step 1.
\[\overrightarrow {OC}\] = \[\overrightarrow {OB}\] + \[\overrightarrow {BC}\]
\[\overrightarrow {OC}\] = ( \[\vec {A_1}\] + \[\vec {A_2}\]) + \[\vec {A_3}\]

Fig. (b)

Step 3: Add \[\vec {A_4}\] to the result from Step 2.
\[\overrightarrow {OD}\] = \[\overrightarrow {OC}\] + \[\overrightarrow {CD}\]
\[\overrightarrow {OD}\] = (\[\vec {A_1}\] + \[\vec {A_2}\] + \[\vec {A_3}\]) + \[\vec {A_4}\]
​The final resultant of all four vectors is the vector \[\overrightarrow {OD}\].