CROSS PRODUCT (Vector Product) Learning Objectives 1). to Use

CROSS PRODUCT (Vector Product)

Learning Objectives

1). To use cross products to: i) calculate the angle ( ) between two vectors and, ii) determine a vector normal to a plane.

2). To do an engineering estimate of these quantities.

Purpose Four primary uses of the cross product are to: 1) calculate the angle ( ) between two vectors, 2) determine a vector normal to a plane, 3) calculate the moment of a force about a point, and 4) calculate the moment of a force about a line.

Definition

|A B| |B A| A B sin θ

In words, |A B| = the component of B perpendicular to A multiplied by the magnitude of ; or vice versa.

Angle ( ) Between Two Vectors

|A×B| sin-1 |A| |B|

Unit Normal Vector

A×B Unit Normal |A×B|

Mechanics (assuming a right-handed coordinate system)

A B (Axi Ayi Azk) (Bxi By j Bzk)

A B (AyBz ByAz ) i (AxBz BxAz ) j (AxBy BxAy ) k

Recall, i i j j k k 0 i k j k i j i j k j i k j k i k j i

Note: Given C A B, C is perpendicular to the plane containing vectors A and B.

Basic Properties 1). A B B A, Use right hand rule to determine direction of resultant vector.

2). p(A B) (pA) B A (pB) A (B C) (A B) (A C) (A B) C (A C) (B C)

3). If |A B| 0, then θ0 If |A B| |A||B|, then θ 90

4). A A 0 39

MOMENT ABOUT A POINT

Learning Objectives

1). To use cross products to calculate the moment of a force about a point. 3). To do an engineering estimate of this quantity.

Definition Moment About a Point: a measure of the tendency of a force to turn a body to which the force is applied.

|M |=r |F|= r sin θ |F|= r F sin θ o op op

|Mo |=|r op | F =|r op | F sin θ = rop F sin θ

|Mo | = |r op ×F|=|rop | |F| sin θ

|(r i r j) (F i F j)| x y x y

|rxFy ryFx ) k|

Comments 1). Direction of the moment can be determined by the right hand rule. 2). Point P can be any point along the line of action of the force

without altering the resultant moment (M)O . 3). Use trigonometric relationships for calculating the moment about a point for 2-D problems. 4). Use vectors and cross products when calculating the moment about a point for 3-D problems.

Moment about a Point Example 2

Given: Angled bar AB has a 200 lb load applied at B.

Find: a) Estimate the magnitude of the moment about fixed support A.

b) Calculate the moment about fixed support A.

Moment about a Point Example 3

Given: Angled bar AO is loaded by cable AB. The tension in cable AB is TAB = 1.2 kN.

Find:

a) Estimate the magnitude and component origins of moment vector M O .

b) Calculate the moment vector about point O ( ) using rOA .

c) Calculate the moment vector about point O ( ) using rOB .

d) How do the solutions for parts (b) and (c) compare?