MAE206-L04-Ch2.5.Vectors-Vector Cross Product

MAE 206 Engineering Statics Prof. Mary Zadeh Assistant Teaching Professor in Mechanical and Aerospace Engineering Department Fall 2023 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 4 – Prof. Zadeh Lecture 4 Vectors: Force and Position Ch 2.5. Vector Cross Product

Ch. 2.5 Page: 2 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 4 – Prof. Zadeh 1 0 . M o m e n t s o f I n e r t i a 9 . F r i c t i o n 7 . C e n t r o i d s & D i s t r i b u t e d F o r c e S y s t e m s 8 . I n t e r n a l F o r c e s 6 . S t r u c t u r a l A n a l y s i s a n d M a c h i n e s 5 . E q u i l i b r i u m o f b o d i e s 4 . M o m e n t o f a F o r c e 3 . E q u i l i b r i u m o f P a r t i c l e s 2 . V e c t o r s : F o r c e a n d P o s i t i o n 1 . I n t r o d u c t i o n t o S t a t i c s MAE 206 Engineering Statics

Ch. 2.5 Page: 4 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 4 – Prof. Zadeh Review - Vector Dot Product • The dot product between two vectors Ԧ ? and ? is an operation defined as Ԧ ? ⋅ ? = Ԧ ? ? cos 𝜃 = ? ? ? ? + ? ? ? ? + ? ? ? ? ▪ 𝜃 is the angle between lines of action of Ԧ ? and ? where 0° ≤ 𝜃 ≤ 180° ▪ The dot product yields a result that is a scalar , and thus the dot product is sometimes called the scalar product . • Angle between the two vectors is provided by 𝜃 = cos −1 Ԧ ? ⋅ ? Ԧ ? ? = cos −1 ? ? ? ? + ? ? ? ? + ? ? ? ? Ԧ ? ?

Ch. 2.5 Page: 5 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 4 – Prof. Zadeh Lecture 4 Objectives • Defining vector cross product. • Determining the normal direction to a plane. • Determining the area of a parallelogram. • Understanding the scalar triple product.

Ch. 2.5 Page: 7 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 4 – Prof. Zadeh Vector Cross Product • The cross product between two vectors is used to determine: 1. the direction normal to the plane containing two vectors 2. the area of a parallelogram formed by two vectors 3. the moment produced by a force . • The units for the cross product are equal to the product of the units for Ԧ ? and ? . • The cross product has the following properties: • Commutative Property: Ԧ ? × ? = − ? × Ԧ ? • Associative property w.r.t multiplication by a scalar: ? Ԧ ? × ? = ? Ԧ ? × ? = Ԧ ? × ? ? • Distributive property w.r.t vector addition: Ԧ ? + ? × Ԧ ? = Ԧ ? × Ԧ ? + ? × Ԧ ?

Ch. 2.5 Page: 8 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 4 – Prof. Zadeh Evaluation of Cross Product • By arranging vectors Ԧ ? and ? in a matrix, the determinant of the matrix can be evaluated to yield Ԧ ? × ? . Ԧ ? × ? = ??? Ƹ ? Ƹ ? ෠ 𝑘 ? ? ? ? ? ? ? ? ? ? ? ? = Ƹ ? Ƹ ? ෠ 𝑘 ? ? ? ? ? ? ? ? ? ? ? ? Ԧ ? × ? = ? ? ? ? − ? ? ? ? Ƹ? + ? ? ? ? − ? ? ? ? Ƹ ? + ? ? ? ? − ? ? ? ? ෠ 𝑘

Ch. 2.5 Page: 10 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 4 – Prof. Zadeh The Normal Direction to a Plane • The cross product can be used to determine the normal direction Ԧ ? to the plane containing two vectors Ԧ ? and ? by evaluating Ԧ ? = Ԧ ? × ? . • If we want the normal direction to be a unit vector , then we normalize Ԧ ? by evaluating ො? = Ԧ 𝐶 Ԧ 𝐶 . • A normal direction can also be obtained by evaluating ? × Ԧ ? and of course, the vector that is produced has opposite direction to Ԧ ? × ? . • In many problems that call for a normal direction to be computed, either direction can be used, while in some problems it may be desirable or necessary to distinguish between these.

Ch. 2.5 Page: 11 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 4 – Prof. Zadeh Area of a Parallelogram • The magnitude of the cross product gives the area of the parallelogram formed by vectors Ԧ ? and ? arranged tail to tail. • The area of the parallelogram is base height = ?? sin 𝜃 • Hence, the area of the parallelogram is the same as the magnitude of the cross product ?? sin 𝜃 = 5.20 m 2 Area of the triangle = ? ? base height = ? ? ?? 𝒔𝒊𝒏 𝜽

Ch. 2.5 Page: 13 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 4 – Prof. Zadeh Example 2.21 Determination of the Normal Direction to a Plane The geometry of the element below is fully described by the coordinates of the corners (nodes). Determine the unit vector in the direction normal to the surface of the element, and the area of the element.

Ch. 2.5 Page: 14 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 4 – Prof. Zadeh Example 2.21 Solution

Ch. 2.5 Page: 16 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 4 – Prof. Zadeh Example 2.19 Solution