Chapter 2Vector AlgebraReviewDr. Ray KwokSJSUVector Algebra - Dr. Ray KwokVector productsBArr⋅(scalar) (scalar) = scalar, (a)(b) = abe.g. 2(4 kg) = 8 kg(scalar) (vector) = vector, e.g. (scalar) (scalar) = scalare.g. 2(4 kg) = 8 kg(vector) times (vector) = ?can be scalar (scalar product, or dot product)orvector (vector product, or cross product)BArr×Can you add a vector to a scalar?()AkAkrr=()yxyxˆ20ˆ10ˆ4ˆ25+=+Vector Algebra - Dr. Ray Kwok( ) ( )zzyyxxzyxzyxBABABABAzˆByˆBxˆBzˆAyˆAxˆABAzˆyˆzˆxˆ0yˆxˆzˆzˆyˆyˆ1xˆxˆcosABBA++=⋅++⋅++=⋅⋅=⋅==⋅⋅=⋅==⋅θ=⋅rrrrrre.g. yˆxˆ3Byˆ2xˆ2A+=−=rrWhat is A · B ?e.g. What’s the angle between these 2 arrows?The scalar product (dot product)Vector Algebra - Dr. Ray KwokInterpretation - projectionVector Algebra - Dr. Ray KwokExample - WorkxFWrr∆⋅=So, is the uplifting force doing anything at all??(Projection)Vector Algebra - Dr. Ray Kwokyˆzˆxˆxˆyˆzˆzˆxˆyˆyˆxˆzˆxˆzˆyˆzˆyˆxˆzˆzˆyˆyˆ0xˆxˆsinABBA−=×−=×−=×=×=×=××=×==×θ=×rre.g. yˆxˆ3Byˆ2xˆ2A+=−=rrWhat is A x B ?e.g.?? What’s the angle between these 2 arrows?zyxzyxBBBAAAzˆyˆxˆBA ≡×rrright-hand coordinatecyclic permutationThe vector product (cross product)Vector Algebra - Dr. Ray KwokExample – calculate torqueox4 N40o1.5 mΣτo= (1.5)(4)sin(40o)= 3.86 N-m (counter-clockwise) choose “+” = counter-clockwiseFrrrr×=τVector Algebra - Dr. Ray KwokExample – perpendicular Fox4 N40o1.5 mΣτo= (1.5)[4 sin(40o)]= 3.86 N-m (counter-clockwise) ox4 sin(40o)1.5 mVector Algebra - Dr. Ray KwokExample – moment armox4 N40o1 m1.5 mΣτo= (4)[1.5 sin(40o)]= 3.86 N-m (counter-clockwise) ox4 N40o1.5 m1.5 sin(40o)Vector Algebra - Dr. Ray KwokExercise - 1Find:(a)(b)(c)(d)(e)(f)(g) Angle between A and B(h) Find a vector that is perpendicular to A and B ?BBAABABAABBArrrrrrrrrrrr⋅××⋅−+2zyxBzxAˆˆˆ2ˆ4ˆ++=−=rrVector Algebra - Dr. Ray KwokTriple vector product( )zyxzyxzyxCCCBBBAAACBA =×⋅rrrscalar(homework))()()()( CBAACBBCACBArrrrrrrrrrrr××≠⋅−⋅=××vectorVector Algebra - Dr. Ray KwokOCCijjieˆeˆδ=⋅OrthogonalCurvilinear – coordinate surfaces can be curved•Transformation between coordinates•Line, Area & Volume integral in each coordinateCartesian, Cylindrical & Spherical coordinatesOrthogonal Curvilinear CoordinatesVector Algebra - Dr. Ray KwokRectangular Polar (2D)Polar to rectangularx = r cos θy = r sin θRectangular to polarr2= x2+ y2 (Pythagorean theorem) tan θ = y/x (be certain which angle is θ)Vector Algebra - Dr. Ray KwokTransformationVector Algebra - Dr. Ray KwokVector operationsVector Algebra - Dr. Ray KwokHomeworkACBBCACBArrrrrrrrr)()()( ⋅−⋅=××Ch.2 - 3, 5, 10, 12, 13, 15, 20, 23, 26, 30,and prove eqn-2.33Also prove that ))(())(()()(
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