a + b is equal to b + a this show that vector addition is
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Consider two vectors
→
A
and
→
B
in any dimension:
→
A
=
<
A
1
,
A
2
,
...
,
A
n
>
→
B
=
<
B
1
,
B
2
,
...
,
B
n
>
Adding these vectors under the usual rules, we obtain:
→
A
+
→
B
=
<
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
→
A
+
→
B
=
<
B
1
+
A
1
,
B
2
+
A
2
,
...
,
B
n
+
A
n
>
Which is, by definition,
→
B
+
→
A
.
∴
→
A
+
→
B
=
→
B
+
→
A
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