Exercise 50 Page 689

We are given the vectors a = ⟨ x_1,y_1 ⟩ and b = ⟨ x_2,y_2 ⟩, and we need to prove the commutative property of addition. a + b = b + a To prove it, we will use vector addition and the commutative property of addition of real numbers.

a + b

SubstituteII

a= ⟨ x_1,y_1 ⟩, b= ⟨ x_2,y_2 ⟩

⟨ x_1,y_1 ⟩ + ⟨ x_2,y_2 ⟩

▼

Simplify

⟨ a,b⟩+⟨ c,d⟩=⟨ a+c,b+d⟩

⟨ x_1+x_2,y_1+y_2 ⟩

CommutativePropAdd

Commutative Property of Addition

⟨ x_2 + x_1,y_2 + y_1 ⟩

⟨ a± c,b± d⟩=⟨ a,b⟩± ⟨ c,d⟩

⟨ x_2,y_2 ⟩ + ⟨ x_1,y_1 ⟩

SubstituteII

⟨ x_1,y_1 ⟩= a, ⟨ x_2,y_2 ⟩= b

b + a

That way we've proven that a + b = b + a.