ME 201 Engineering Mechanics: Statics Chapter 2 – Part E 2.9 Dot Product

Key Concepts  Dot Product Definition & Properties  Angle between 2 Vectors  Projection of a Vector along an Axis

Dot Product  Dot Product is a scalar operation used in projecting vectors onto a given axis A ∙ B = A B cos θ Where 0º ≤ θ ≤ 180º θ A B

Dot Product Laws of Operation Commutative Law A ∙ B = B ∙ A Multiplication by Scalar a(A ∙ B) = (aA) ∙ B = A ∙ (aB) = (A ∙ B)a Distributive Law A ∙ (B + D) = (A ∙ B) + (A ∙ D)

Dot Product of Cartesian Vector A ∙ B = A B cos θ i ∙ i = (1) (1) cos 0 = 1 i ∙ j = (1) (1) cos 90 = 0 i ∙ k = (1) (1) cos 90 = 0 j ∙ j = (1) (1) cos 0 = 1 j ∙ k = (1) (1) cos 90 = 0 k ∙ k = (1) (1) cos 0 = 1

Dot Product Cartesian Vector Formulation A ∙ B = (A x i +A y j + A z k) ∙ (B x i + B y j + B z k) = A x B x (i ∙ i) + A x B y (i ∙ j) +A x B z (i ∙ k) +A y B x (j ∙ i) + A y B y (j ∙ j) +A y B z (j ∙ k) +A z B x (k ∙ i) + A z B y (k ∙ j) + A z B z (k ∙ k) = A x B x + A y B y + A z B z

Dot Product Cartesian Vector Formulation A ∙ B = A x B x + A y B y + A z B z = A B cos θ

Using the Dot Product to Find the Angle Between 2 Vectors  Given:  Solution:

Projection of a Vector onto a Given Axis  The scalar projection of A along a line is determined from the dot product of A and the unit vector U which defines the direction of the line A ∙ U = A U cos θ = A cos θ = A U θ A UAUAU

Using the Dot Product to Find the Projection of a Vector onto a Given Axis  Given:  Solution: