MAE206-L03-Ch2.3-4.Vectors-Cartesian Representation of Vectors in 3-D Vector Dot 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 3 – Prof. Zadeh Lecture 3 Vectors: Force and Position Ch 2.3. Cartesian Representation of Vectors in 3-D Ch 2.4. Vector Dot Product

Ch. 2.3 Page: 2 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – 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. 2.3 Page: 4 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – Prof. Zadeh Review • 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. 2.3 Page: 5 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – Prof. Zadeh Review - 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. 2.3 Page: 7 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – Prof. Zadeh Lecture 3 Objectives Chapter 2.3 Understand Right-hand Cartesian coordinate system. Chapter 2.4 Find the vector dot product of two vectors. Calculate the angle between two vectors.

Ch. 2.3 Page: 8 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – Prof. Zadeh Right-Hand Cartesian Coordinate System • Three dimensions Cartesian coordinate system uses three orthogonal reference directions consisting of x , y , and z directions. • Proper interpretation of many vector operations, such as the cross product , requires the x , y , and z directions arranged in a consistent manner. • The universal convention in mechanics and vector mathematics in general is z must be taken in the direction shown in Figure and the result is called a right-hand coordinate system .

Ch. 2.3 Page: 10 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – Prof. Zadeh Direction Angles • An effective way to characterize a vector’s orientation is to use direction angles . • Direction angles 𝜃 ? , 𝜃 ? , 𝜃 ? and are defined to be the angles measured from the positive x , y , and z directions, respectively, to the direction of the vector. • Direction angles have the values between 0° and 180°. • Direction angles can be used to obtain vector’s components, and vice versa.

Ch. 2.3 Page: 11 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – Prof. Zadeh Direction Angles • Consider a vector polygon shown in the figure. This polygon is a right triangle that consists of the vector’s magnitude Ԧ? . Using elementary trigonometry: ? ? = Ԧ? 𝑐𝑜?𝜃 ? ? ? = Ԧ? 𝑐𝑜?𝜃 ? ? ? = Ԧ? 𝑐𝑜?𝜃 ? • Hence, Ԧ? = ? ? Ƹ ? + ? ? Ƹ ? + ? ? ෠ 𝑘 Ԧ? = Ԧ? cos 𝜃 ? Ƹ ? + Ԧ? cos 𝜃 ? Ƹ ? + Ԧ? cos 𝜃 ? ෠ 𝑘 Ԧ? = Ԧ? cos 𝜃 ? Ƹ ? + cos 𝜃 ? Ƹ ? + cos 𝜃 ? ෠ 𝑘 Eqn. (2.25) unit vector that points in the direction of 𝒗

Ch. 2.3 Page: 13 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – Prof. Zadeh Position Vectors • Construction of a position vector in three dimensions is analogous to the procedure described in the previous section. • Given points T and H have the coordinates ? 𝑇 , ? 𝑇 , ? 𝑇 and ? 𝐻 , ? 𝐻 , ? 𝐻 . • The position vector from the tail T to head H is denoted by Ԧ ? 𝑇𝐻 = ? 𝐻 − ? 𝑇 Ƹ ? + ? 𝐻 − ? 𝑇 Ƹ ? + ? 𝐻 − ? 𝑇 ෠ 𝑘

Ch. 2.3 Page: 14 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – Prof. Zadeh Use of Position Vectors to Write Expressions for Force Vectors • If a force vector Ԧ 𝐹 lies along the line of action of a position vector Ԧ ? , or is know to be parallel to the position vector, then a vector expression for the force may be written as Ԧ 𝐹 = 𝐹 ො? 𝑟 = 𝐹 Ԧ 𝑟 Ԧ 𝑟 where Ԧ 𝑟 Ԧ 𝑟 is a unit vector that gives Ԧ 𝐹 proper direction, and F is the component of the force Ԧ 𝐹 along the direction Ԧ ? . • Often, the location of two points on the line of action of a force will be known, and Ԧ ? can be obtained by taking the difference between coordinates of the position vector’s head and tail . • Note that in an expression such as Ԧ 𝐹 = 𝐹 Ԧ 𝑟 Ԧ 𝑟 ▪ if F > 0, then the direction of the force is the same as direction of Ԧ ? . ▪ if F < 0, then the direction of the force is opposite direction of Ԧ ? .

Ch. 2.3 Page: 16 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – Prof. Zadeh Example 2.10 Position Vectors and Force Vectors Two cables apply forces to the cantilever I beam. Cable AB applies a tensile force of magnitude 𝑃 = 2 kN , and cable CD applies a tensile force of magnitude 𝐹 = 1 kN . Write expressions for the forces 𝑊 , Ԧ 𝐹 , and 𝑃 applied to the I beam.

Ch. 2.3 Page: 17 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – Prof. Zadeh Example 2.14 Solution

Ch. 2.4 Page: 19 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – Prof. Zadeh Dot Product using Cartesian Components • The dot products between combinations of unit vectors are • The dot product is between the vector Ԧ ? ⋅ ? = ? ? ? ? + ? ? ? ? + ? ? ? ? • Angle between the two vectors is provided by 𝜃 = cos −1 Ԧ ? ⋅ ? Ԧ ? ? = cos −1 ? ? ? ? + ? ? ? ? + ? ? ? ? Ԧ ? ? Ƹ? ⋅ Ƹ? = 𝟏 Ƹ ? ⋅ Ƹ ? = 0 Ƹ ? ⋅ ෠ 𝑘 = 0 Ƹ ? ⋅ Ƹ ? = 0 Ƹ ? ⋅ Ƹ ? = 𝟏 Ƹ ? ⋅ ෠ 𝑘 = 0 ෠ 𝑘 ⋅ Ƹ ? = 0 ෠ 𝑘 ⋅ Ƹ ? = 0 ෡ 𝒌 ⋅ ෡ 𝒌 = 𝟏

Ch. 2.4 Page: 20 Let’s GO PACK MAE 206 – Fall 2023 – Lecture 3 – Prof. Zadeh Component of a Vector in a Particular Direction • Consider two vectors Ԧ 𝐹 and Ԧ ? and arrange them tail to tail, which leads to Figure below. The parallel component of Ԧ 𝐹 or in other words, the amount of Ԧ 𝐹 that acts in direction Ԧ ? , is denoted by 𝐹 ∥ where the subscript ∥ means parallel to 𝒓 and is given by 𝐹 ∥ = Ԧ 𝐹 ⋅ Ԧ ? Ԧ ? • Basic trigonometry gives us the component of Ԧ 𝐹 in the direction of Ԧ ? as 𝐹 ∥ = Ԧ 𝐹 cos 𝜃