PHYS 218sec. 517-520ReviewChap. 1Caution• This presentation is to help you understand the contents of the textbook.• Do not rely on this review for preparing exams.• Solving and understanding all exercises and examples are very important. Also, try to solve the end of chapter problems.Chap. 1• Unit: International System (SI unit)– Length: meter (m)– Mass: kilogram (kg)– Time: second (s)• Unit prefixes• See pps file for recitation (Sep. 1)Vector • Scalar: a quantity which can be described by a single numberVector: a quantity which has a magnitude and a direction– Mathematically, this is not the correct definition, but is enough for our purpose.– Notation: – Magnitude of a vector:– Be careful with the notations when you read other books or articles. • A scalar can be multiplied to a vector– Changes the magnitude and the direction of the vector–E.g.• When two vectors have the same magnitude and the same direction, they are identical. or A AGG or orAAAGG has the same magnitude as but has the opposite direction.−AAGGVector Addition: commutative=+=+CABBAGG GGGAGBGCGTo add two vectors in a graphical way, place the tail of the second vector at the head of the first vector, i.e., the starting point of the first vector becomes the starting point of the sum and the ending point of the second vector becomes the ending point of the sum.Vector SubtractionSubtracting vector B from vector A is identical to add vector A and vector (-B).()= − =+−CABA BGG GGGIf you are not sure with the direction of , note that = + .CABCGGGGDimension of space1-dimension, one number is enough to specify the position.0xxy2-dimension, two numbers (x,y) are needed to specify the position.xzy3-dimension, three numbers (x,y,z) are needed to specify the position.Unit vectorA vector which has a magnitude of 1. It only specify the direction.ˆnotation: the unit vector in the direction of →AAGCoordinate systemCartesian (or rectangular) coordinate system2-dim.3-dim.ˆiˆjˆˆˆ, , + x,+ y,and are unit vectors pointing directions, respecti(basi+ s velvec ry.s)ztoijk1ˆiˆjˆkComponents of vectorsYou can write any vector as a linear combination of basis vectors.()ˆˆ or you can write ,, are called the components of xyx y xyxyAAAAAA=+=+ =AA A i j AAGG G GGPolar coordinate systemAGxAGyAGYou can also use the magnitude (A) and the direction (θ) of a vector for writing the vector.AGθARelations between (Ax,Ay) and (A,θ)()y22xAIf you know , , and =arctan .Axy x yAA A A A θ⎛⎞⎟⎜⎟=+⎜⎟⎜⎟⎜⎝⎠()If you know , , cos , sin .xyAAA AAθθθ==θ is measured from the +x axis, rotating toward +y-axisThis equation gives two solutions for θ. Draw diagramsto see which is thecorrect answer.In 3-dimensions222xyzAAAA=++In polar coordinates, we have (A,θ,φ). But we do not discuss it.Products of vectorsScalar (dot, inner) productThis gives a scalar quantity.Notation: ⋅ABGGDefinition: cos , where is the angle between and .AB φφ⋅ =AB A BGGGGθ0<θ<π2: this gives the squared magnitude of .A⋅ =AA AGG G0, if 0< ,20, if <2πφπφπ⋅ ><⋅ <<ABABGGGGScalar product using componentsUnit vectors: have magnitude 1 and perpendicular to each otherˆˆ ˆˆ ˆˆ ˆ ˆˆˆ ˆ ˆ1, 0⋅ = ⋅ = ⋅ = ⋅ = ⋅ = ⋅ =ii jj kk ij jk ki2ˆˆˆIn 3-dim., if , thenˆˆ ˆˆˆˆ()()ˆˆ ˆˆxyzxyz xyzxxyAAAAAA AAAAAA=++⋅ =++⋅ ++= ⋅ + ⋅AijkAA i j k i j kii ijGGGˆˆxzAA+ ⋅ikˆˆyxAA+ ⋅ji2ˆˆ ˆˆyyzAAA+ ⋅ + ⋅jjjkˆˆzxAA+ ⋅kiˆˆzyAA+ ⋅kj2222ˆˆzxyzAAAA+ ⋅=++kkˆˆ ˆˆˆˆFor two vectors, and ˆˆ ˆˆˆˆ()()xyz xyzxyz xyzxx yy zzAAA BBBAAA BBBAB AB AB=++ =++⋅ =++ ⋅ ++=++AijkBijkAB i j k i j kGGGGThis is a general expression. When the vectors are written or given with their components, you can always use this relation. You can use itas the definition of scalar product. But remember that this relation holdsonly when the basis vectors (i, j, k) are normal to each other.Vector (cross, outer) productThe vector product of two vectors gives another vector.Notation: ×ABGG of : sin , where is the angle between the two vectors (0 )Therefore, the vector product of two (anti)parallel vectors gives a null vector ,since =0 or and sin(0) sinMagnitude AB φφφπφπ××=≤≤=AB AB0GGGGG() 0.φ =of : given by the right-hand ruleTherefore, is always perpendicular to the plane madDirecte by ion()( an)d 0⋅ ×=⋅ ×=××⇒ABABABAAB BABGGGGGGGG GGG G ×=−∴ ×AB BAGGGGThe vector product is NOT commutative.Vector product using componentsWe use the right-handed coordinate system.ˆiˆjˆkThe vector product isdefined for 3-dim.vectors.For two vectors, ˆˆ ˆˆˆˆ and ˆˆ ˆˆˆˆ()()ˆˆˆ()()()xyz xyzxyz xyzyz zy zx xz xy yxAAA BBBAAA BBBABAB ABAB ABAB=++ =++×= + + × + += − + − + −A i jkB i jkAB i j k i j kijkGGGGSo we haveˆˆ ˆˆˆˆ ˆˆ,ˆˆ ˆˆ,ˆˆ ˆ ˆ,×=×=×=×= ×=−×= ×=−×= × =−ii jj kk 0ijk ji kjki kj iki j ik jGGGGGGGGGIf they are in cyclic order, then you have (+) sign.If not, you have (-) sign.Cyclic order(x y z)J(y z x)J(z x y)or (i j k) or (1 2 3)xyzIf you know how to calculate the determinant of a 3x3 matrix ˆˆˆˆˆˆ()()()xy z yz zy zx xz xy yxxyzAA A AB AB AB AB AB ABBBB×= = − + − + −ijkAB i j kGGThis is a coincidence. Matrix has no relation with the vector product.But if you are familiar with the matrix determinant, this is a goodway to memorize the components of a vector
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