Exercise 51 Page 689

We are given the vectors a = ⟨ x_1,y_1 ⟩ and b = ⟨ x_2,y_2 ⟩, a scalar k, and we need to prove the property written below. k(a + b) = ka + kb To prove it, we will use the scalar multiplication of vectors which states that k⟨ x,y ⟩ = ⟨ kx , ky ⟩.

k(a + b)

SubstituteII

a= ⟨ x_1,y_1 ⟩, b= ⟨ x_2,y_2 ⟩

k( ⟨ x_1,y_1 ⟩ + ⟨ x_2,y_2 ⟩)

▼

Simplify

⟨ a,b⟩+⟨ c,d⟩=⟨ a+c,b+d⟩

k⟨ x_1+x_2,y_1+y_2 ⟩

Distribute the scalar

⟨ k(x_1+x_2),k(y_1+y_2) ⟩

Distr

Distribute k

⟨ kx_1 + kx_2,ky_1 + ky_2 ⟩

⟨ a± c,b± d⟩=⟨ a,b⟩± ⟨ c,d⟩

⟨ kx_1,ky_1 ⟩ + ⟨ kx_2,ky_2 ⟩

Factor out the scalar

k⟨ x_1,y_1 ⟩ + k⟨ x_2,y_2 ⟩

SubstituteII

⟨ x_1,y_1 ⟩= a, ⟨ x_2,y_2 ⟩= b

k a + k b