𝑣=⟨7,3,10⟩ v = ⟨ 7 , 3 , 10 ⟩ 𝑤=⟨6,6,10⟩ w = ⟨ 6 , 6 , 10 ⟩ Find the cosine of the angle between 𝑣 v and 𝑤 w .

Question

Find the cosine of the angle between  and .

Answer 1

Let v = <7, 3, 10> and w = <6, 6, 10>. The magnitude of v is |v| = √(7² + 3² + 10²) = √158 and the magnitude of w is |w| = √(6² + 6² + 10²) = √172 = 2√43.

There are two definitions of the dot product that we are going to use. The first is that the dot product of two vectors is the sum of the product of their respective components. In algebraic form,

v • w = ∑v_iw_i

v • w = 7*6 + 3*6 + 10*10

v • w = 42 + 18 + 100

v • w = 160

The second definition of the dot product is that the dot product of two vectors is the product of their magnitudes, multiplied by the cosine of the angle between them. Let the angle between v and w be θ such that

v • w = |v|*|w|*cos(θ)

From this definition, we can express cos(θ) as

cos(θ) = (v • w)/(|v|*|w|)

Since we have the dot product and the magnitudes, we can simply substitute:

cos(θ) = (160)/((√158)*(2√43))

cos(θ) = 80/√6794

cos(θ) ≈ 0.9706

From this, we can see that the angle between the two vectors is close to 0, as we'd expect by looking at the vectors themselves.