Show that addition of vectors is commutative, i.e. A+B=B+A?

1 Answer
Morgan
Dec 30, 2017

See below.

Explanation:

Consider two vectors #vecA# and #vecB# in any dimension:

#vecA= < A_1,A_2,...,A_n > #

#vecB= < B_1,B_2,...,B_n > #

Adding these vectors under the usual rules, we obtain:

#vecA+vecB= < A_1+B_1, A_2 + B_2,...,A_n+B_n > #

But each component of a vector is just a real number, and we know that real numbers are commutative. Therefore, using the commutative property of real numbers under addition, we may equivalently write

#vecA+vecB= < B_1+A_1,B_2 + A_2,...,B_n+A_n > #

Which is, by definition, #vecB+vecA#.

#:.vecA+vecB=vecB+vecA#

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