Class 11 > Unit # 02: Kinematics > Properties of Addition of Vectors

Class 11 Physics • Unit 2 Kinematics

Properties of Addition of Vectors

Detailed study of the Commutative and Associative laws of vector addition using geometric proofs and the head-to-tail rule.

1. Commutative Law of Vector Addition

Commutative Property: The order in which two vectors are added does not change the net resultant vector. Changing vector sequencing yields the exact same magnitude and direction.

$$\vec{A} + \vec{B} = \vec{B} + \vec{A}$$

Geometric Verification via Parallelogram Method:

Case 1: $\vec{A} + \vec{B}$

• Draw vector $\vec{A}$ first.
• Attach the tail of vector $\vec{B}$ to the head of $\vec{A}$.
• The resultant vector $\vec{R}$ spans from the tail of $\vec{A}$ to the head of $\vec{B}$.

Case 2: $\vec{B} + \vec{A}$

• Draw vector $\vec{B}$ first.
• Attach the tail of vector $\vec{A}$ to the head of $\vec{B}$.
• The resultant vector $\vec{R}$ spans from the tail of $\vec{B}$ to the head of $\vec{A}$.

Key Insight: Both geometric constructions form a parallelogram where both pathways share the exact same diagonal representing $\vec{R}$, proving that vector addition is strictly commutative.

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2. Associative Law of Vector Addition

Associative Property: When adding three vectors, the grouping or order of combination does not alter the final resultant vector.

$$(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$$

Geometric Verification Step-by-Step:

1. Left-Hand Side Grouping $[(\vec{A} + \vec{B}) + \vec{C}]$:
   • First find intermediate resultant $\vec{R}_1 = \vec{A} + \vec{B}$ using the head-to-tail method.
   • Add vector $\vec{C}$ to the head of $\vec{R}_1$ to derive the final resultant $\vec{R}$.
2. Right-Hand Side Grouping $[\vec{A} + (\vec{B} + \vec{C})]$:
   • First find intermediate resultant $\vec{R}_2 = \vec{B} + \vec{C}$ using the head-to-tail method.
   • Add vector $\vec{A}$ to the tail of $\vec{R}_2$ to derive the final resultant $\vec{R}$.

Conclusion: Since both grouping configurations result in the identical final vector $\vec{R}$, vector addition satisfies the associative law, confirming that multiple vectors can be added in any arbitrary grouping order.