Lecture 03 - Lecture 03 * CARTESIAN VECTORS & ADDITION...

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Statics: Lecture Notes for Sections 2.5-2.9 1 Lecture 03 * CARTESIAN VECTORS & ADDITION & SUBTRACTION OF CARTESIAN VECTORS * POSITION VECTORS & FORCE VECTORS * DOT PRODUCT DOT PRODUCT Section 2.5-2.9 CARTESIAN VECTORS & ADDITION & SUBTRACTION OF CARTESIAN VECTORS Objectives : Students will be able to : a) Represent a 3-D vector in a Cartesian coordinate system. b) Find the magnitude and coordinate angles of a 3-D vector c) Add vectors (forces) in 3-D space 2
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Statics: Lecture Notes for Sections 2.5-2.9 2 APPLICATIONS Many problems in rea life Many problems in real-life involve 3-Dimensional Space. How will you represent each of the cable forces in Cartesian vector form? 3 APPLICATIONS (continued) Given the forces in the cables, How will you determine the resultant force acting at D the top of How will you determine the resultant force acting at D, the top of the tower ? 4
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Statics: Lecture Notes for Sections 2.5-2.9 3 Right Handed Coordinate System Z X 5 Y Z Rectangular Components z y x k A A A A A A A A A A z A z y x z y x A A A A j i 6 X Y x A y A’ Orthorhombic
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Statics: Lecture Notes for Sections 2.5-2.9 4 A UNIT VECTOR For a vector A with a magnitude of A, an unit vector is defined as U A = A / A . Characteristics of a unit vector : a) Its magnitude is 1. b) It is dimensionless . c) It points in the same direction as the original vector ( A ). 7 The unit vectors in the Cartesian axis system are i , j , and k . They are unit vectors along the positive x, y, and z axes respectively. Unit Direction Vector A A A A u A u A A u A A 8 1
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Statics: Lecture Notes for Sections 2.5-2.9 5 How to calculate a unit vector 2 2 2 2 2 2 y x y z y x x z y x A A A A A A A A A A A A A u 2 2 2 y x z y x z y x A A A A A A A A A A A u 9 2 2 2 z y x z z A A A A z Cartesian Unit Vectors Z Magnitude j k Direction Point of Application Sense 10 X Y i
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Statics: Lecture Notes for Sections 2.5-2.9 6 Magnitude of a Cartesian Vector z y x k A j A i A A z y x A A A A 11 2 2 2 z y x A A A A A Direction of a Cartesian Vector Z x = (alpha) A j A y k A z y = (beta) z = (gamma) 12 X Y i A x
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Statics: Lecture Notes for Sections 2.5-2.9 7 Direction Cosines Z x = (alpha) A A 13 X Y i A x i A x A A cos x Direction Cosines Z y = (beta) A A 14 X Y j A y j A y A A cos y
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Statics: Lecture Notes for Sections 2.5-2.9 8 Direction Cosines Z z = (gamma) A k A z k A z A 15 X Y A A z cos 3-D CARTESIAN VECTOR TERMINOLOGY C id b ith id A
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Lecture 03 - Lecture 03 * CARTESIAN VECTORS & ADDITION...

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