Engr Help

Vectors – Dot Product

Definition of Dot Product of Two Vectors U = (u₁,u₂,u₃) and V = (v₁, v₂, v₃)

U ∙ V = (u₁ v₁ + u₂v₂+u₃v₃) The result of the dot product is a scalar value.

Interpretation of Dot Product Let θ be the angle between U and V .

U ∙ V = |U| |V| cos θ

So to calculate the angle between two vectors use: cos θ = U ∙ V / |U| |V|

Example of a Dot Product

Note: i ∙ i = 1 , j ∙ j = 1 , k ∙ k = 1 , i ∙ j = 0, i ∙ k = 0, j ∙ k = 0

Let U = 3 i + 4 j + 5 k and V = - i + j - 6 k

| U | = √ ( 3² + 4² + 5²) = √ 50 and | V | = √ (1² + 1² + 6² ) = √ 38

U ∙ V = (3)(- 1) + (4)(1) + (5)(- 6) = - 31 (scalar result)

And the angle θ between the two vectors is from cos θ = - 31 / [( √ 50 )(√ 38 )]