Vectors – Addition, Dot Product, Direction Cosines

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

U ∙ V = (u₁ v₁ + u₂ v₂ + u₃ v₃) 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 cos θ = U ∙ V / |U| |V

One can use this dot product to calculate the angle between a vector U and each

coordinate axis, x, y, and z. Call them θ_x, θ_y, θ_z . Then the cosine of these angles

are called the “direction cosines” . i.e.

cosθ_x = U ∙ i / |U| |i| = u₁ / |U| , cosθ_y = U ∙ j / |U| |j| = u₂ / |U|

cosθ_z = U ∙ k / |U| |k| = u₃ / |U|

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

Click here to continue with discussion on scalar and vector projections.

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