Commutative

Definition

Commutative refers to the property of an operation where changing the order of the operands does not change the result. In vector addition, for example, $\vec{A} + \vec{B} = \vec{B} + \vec{A}$.

5 Must Know Facts For Your Next Test

1. Vector addition is commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$.
2. Scalar multiplication is also commutative: $a \cdot b = b \cdot a$.
3. The commutative property does not apply to vector subtraction: $\vec{A} - \vec{B} \neq \vec{B} - \vec{A}$ in general.
4. In physics, understanding that forces and displacements are vectors helps in applying the commutative property correctly during problem-solving.
5. Many physical laws rely on vector addition being commutative, ensuring that different paths yield the same resultant vector.

Review Questions

• What is an example of a commutative operation with vectors?
• Does the commutative property apply to vector subtraction? Why or why not?
• How does understanding the commutative property assist in solving physics problems involving vectors?

Related terms

Associative: The associative property states that how you group numbers or vectors when adding or multiplying does not change their result: $(\vec{A} + (\vec{B} + \vec{C})) = ((\vec{A} + \vec{B}) + \vec{C})$.

Distributive: $a(\vec{b} + \vec{c}) = a\cdot\vec {b}+a\cdot\ vec {c}$ illustrates how multiplication distributes over addition whether dealing with scalars or vectors.

Scalar: $a$ scalar is a quantity described by magnitude only and follows standard arithmetic rules including commutativity.