MAE206-L02-Ch2.1-2.Vectors-Basic Concept Cartesian Representation of Vectors in 2-D

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 2 – Prof. Zadeh Lecture 2 Vectors: Force and Position Ch 2.1. Basic Concepts Ch 2.2. Cartesian Representation of Vectors in 2-D

Ch. 2.1 Page: 2 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – 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. 2.1 Page: 4 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – Prof. Zadeh Objectives • Performing basic vector operations like addition, subtraction, and multiplying a vector by a scalar. • Resolving vectors into components. • Defining unit vectors. • Defining Cartesian coordinate system. • Performing vector operations using Cartesian components.

Ch. 2.1 Page: 5 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – Prof. Zadeh Introduction • A vector has both size and direction. • A vector has a head and a tail , as well as a line of action , which is the line of infinite extent along which the vector is positioned. • The magnitude of a vector is the measure of its size, or length, including appropriate units. ▪ The magnitude of a vector is a positive scalar of any vector that has nonzero size, and is zero only for a vector of zero size. The magnitude can never be negative.

Ch. 2.1 Page: 7 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – Prof. Zadeh Polar Vector Representation • A simple representation we will sometimes use for writing vectors in two dimensions is called polar vector representation . • This representation consist of a statement of the vector’s magnitude and direction referred to a right-hand horizontal reference direction, with a positive angle being measured counterclockwise.

Ch. 2.1 Page: 8 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – Prof. Zadeh Basic Vector Operations • Equivalent vectors: Two vectors are said to be equivalent , or equal, if they have the same magnitude and orientation . ▪ Note that two equivalent vectors may have different lines of action, provided the lines of action are parallel . • Vector addition: Addition of two vectors Ԧ ? and ? produces a new vector 𝑅 and this operation is denoted by 𝑅 = Ԧ ? + ? . ▪ Vector addition has the following properties: • commutative property: Ԧ ? + ? = ? + Ԧ ? • associative property: Ԧ ? + ? + Ԧ ? = Ԧ ? + ? + Ԧ ?

Ch. 2.1 Page: 10 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – Prof. Zadeh Vector Subtraction • Subtracting a vector ? from a vector Ԧ ? is denoted by Ԧ ? − ? and is defined as: Ԧ ? − ? = Ԧ ? + (−1) ? Comparison of addition and subtraction of two vectors

Ch. 2.1 Page: 11 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – Prof. Zadeh Performing Vector Operations • Since the addition of two vectors will generally involve a triangle or parallelogram of complex geometry, the laws of sines and cosines will be useful. • For a general triangle ▪ Law of sines sin𝜃 ? ? = sin𝜃 ? ? = sin𝜃 ? ? ▪ Law of cosines ? = ? 2 + ? 2 − 2?? cos𝜃 ? ? = ? 2 + ? 2 − 2?? cos𝜃 ? ? = ? 2 + ? 2 − 2?? cos𝜃 ? • For a right triangle • ▪ In the special case of a right triangle, the laws of sines and cosines simplify to familiar expressions ? = ? cos𝜃 ? = ? sin𝜃 ? ? = ? sin𝜃 ? = ? cos𝜃 ? ? = ? 2 + ? 2 ▪ Pythagorean theorem 𝒔𝒊?𝜽 ? = ? ? & ??𝒔𝜽 ? = ? ? 𝒔𝒊?𝜽 ? = ? ? & ??𝒔𝜽 ? = ? ?

Ch. 2.1 Page: 13 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – Prof. Zadeh Example 2.1 Addition of Vectors A D ring is sewn on a backpack for use in securing miscellaneous items to the outside of the backpack. If the D ring has three cords tied to it and the cords support the forces shown, determine the resultant force applied to the D ring by the cords, expressing the result as a vector.

Ch. 2.1 Page: 14 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – Prof. Zadeh Example 2.1 Solution

Ch. 2.2 Page: 16 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – Prof. Zadeh Unit Vector • A unit vector is defined to be a dimensionless vector that has unit magnitude . • Given any arbitrary vector having nonzero magnitude, we may construct a unit vector that has the same direction, using ො? = Ԧ? Ԧ? • We will use a “hat” symbol ( ෡ ) over unit vectors, whereas all other vectors will use an arrow symbol ( ). • The vector Ԧ? and its magnitude Ԧ? have the same units, thus unit vector ො? is dimensionless.

Ch. 2.2 Page: 17 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – Prof. Zadeh Cartesian Coordinate System • In two dimensions a Cartesian coordinate system uses two orthogonal reference directions, which we will usually call the ? and ? directions. • The intersection of the reference direction is the origin of the coordinate system, and any point in the ?? plane is identified by its ? and ? coordinates. • We denote the coordinates of a point P using an ordered pair of numbers ( ? 𝑃 and ? 𝑃 ) where and are the ? and ? coordinates of the point, respectively, measured from the origin.

Ch. 2.2 Page: 19 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – Prof. Zadeh Addition of Vectors using Cartesian Components • Consider the addition of the two vectors in figure (a). We first write the vectors using Cartesian representation. Thus, Ԧ? 1 = ? 1? Ƹ ? + ? 1? Ƹ ? Ԧ? 2 = ? 2? Ƹ ? + ? 2? Ƹ ? • The resultant vector is the sum of Ԧ? 1 and Ԧ? 2 : 𝑅 = Ԧ? 1 + Ԧ? 2 𝑅 = ? 1? Ƹ ? + ? 1? Ƹ ? + ? 2? Ƹ ? + ? 2? Ƹ ? 𝑅 = ? 1? + ? 2? Ƹ ? + ? 1? + ? 2? Ƹ ?

Ch. 2.2 Page: 20 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 2 – Prof. Zadeh Position Vectors • The spatial position of one point (head of vector) relative to another point (tail of vector) is provided by a position vector . • For example, in figure below, the two points T and H denote the tail and the head of a position. The position vector from T to H is denoted by Ԧ ? 𝑇𝐻 which is given as Ԧ ? 𝑇𝐻 = ? 𝐻 − ? 𝑇 Ƹ ? + ? 𝐻 − ? 𝑇 Ƹ ?