PowerPoint PresentationHomework ProblemsSlide 3Slide 43-D Vectors; Base Vectors3-D Rectangular CoordinatesWriting 3-D Components3-D Direction CosinesUnit Base VectorsVector Equality in ComponentsAdditional Vector OperationsGeneral Unit VectorsPosition Vectors in SpaceVectors in Matrix FormSummaryExample ProblemME 221 Lecture 4 1ME 221 StaticsLecture #4Sections 2.4 – 2.5ME 221 Lecture 4 2Homework Problems•Due Today:–1.1, 1.3, 1.4, 1.6, 1.7–2.1, 2.2, 2.11, 2.15, 2.21•Due Wednesday, September 9:–Chapt 2: 22, 23, 25, 27, 29, 32, 37, 45, 47 & 50–On 2.50:•Solve with hand calculations first•Then use MathCAD, MatLab, Excel, etc. to solve•Quiz #1 – Friday, 9/5ME 221 Lecture 4 3TA Hours• Help Sessions – ME Help Room – 1522EB - Cubicle #2 • TA’s: – Jimmy Issa, Nanda Methil-Sudhakaran & Steve RundellMondays & Wednesdays – 10:15am to 5:00pmTuesdays & Thursdays – 8:00am to 5:00pmFridays – 8:00am to 11:00am• Grader – Jagadish Gattu2415EB – Weds: 10:00am to 12:00amME 221 Lecture 4 4Last Lecture•Vector Components• Scalar Multiplication of Vectors• Perpendicular Vectors• Example 2.3ME 221 Lecture 4 53-D Vectors; Base Vectors•Rectangular Cartesian coordinates (3-D)•Unit base vectors (2-D and 3-D)•Arbitrary unit vectors•Example problem•Vector component manipulationME 221 Lecture 4 63-D Rectangular Coordinates•Coordinate axes are defined by OxyzxyzOCoordinates can be rotatedany way we like, but ...•Coordinate axes must be a right-handed coordinate system.ME 221 Lecture 4 7xyzOA =Writing 3-D Components•Component vectors add to give the vector:xyzOA=AxAx +AyAy +AzAzAlso,2x 2 2y zA A A AME 221 Lecture 4 83-D Direction CosinesThe angle between the vector and coordinate axis measured in the plane of the twoxyzOAxcosx x xAAycosyy y AAzcoszz z AAWhere: x2+y2+z2=1ME 221 Lecture 4 9Unit Base VectorsAssociate with each coordinate, x, y, and z, a unit vector (“hat”). All component calculations use the unit base vectors as grouping vectors.xyzOˆˆjˆkNow write vector as follows:ˆˆˆx y zA A j A k Awhere Ax = |Ax|Ay = |Ay|Az = |Az|ME 221 Lecture 4 10Vector Equality in Components•Two vectors are equal if they have equal components when referred to the same reference frame. That is:ˆ ˆˆ ˆˆ ˆx y z x y zA A j A k B B j B k ifAx = Bx , Ay = By , Az = BzME 221 Lecture 4 11Additional Vector Operations•To add vectors, simply group base vectors ˆ ˆˆ ˆˆ ˆˆˆˆx y z x y zx x y y z zA A j A k B B j B kA B A B j A B k •A scalar times vector A simply scales all the components ˆ ˆˆ ˆˆ ˆx y z x y zA A j A k A A j A k ME 221 Lecture 4 12General Unit Vectors•Any vector divided by its magnitude forms a unit vector in the direction of the vector.–Again we use “hats” to designate unit vectorˆˆˆˆyxzbbbj k bbb b b bxyzObˆbME 221 Lecture 4 13Position Vectors in Space•Points A and B in space are referred to in terms of their position vectors.xyzOrAˆˆˆA A A Ax y j z k rrBˆˆˆB B B Bx y j z k rrB/A /ˆˆˆB BA AB A B Aj z ky zx x y r•Relative position defined by the differenceME 221 Lecture 4 14Vectors in Matrix Form•When using MathCAD or setting up a system of equations, we will write vectors in a matrix form:ˆˆˆxx y z yzAA A j A k AA AME 221 Lecture 4 15Summary•Write vector components in terms of base vectors•Know how to generate a 3-D unit vector from any given vectorME 221 Lecture 4 16Example
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