Lecture 3: 6/26/2003, CROSS PRODUCT Maths21a, O. KnillCROSS PRODUCT. The cross product of two vectors~v = (v1, v2, v3) and ~w = (w1, w2, w3) is defined as the vector~v × ~w = (v2w3− v3w2, v3w1− v1w3, v1w2− v2w1).To compute it:multiply diagonallyat the crosses.v1v2v3v1v2X X Xw1w2w3w1w2DIRECTION OF ~v × ~w: ~v × ~w is orthogonal to ~v and orthogonal to ~w.Proof. Check that ~v · (~v × ~w) = 0.LENGTH:|~v × ~w| = |~v|| ~w| sin(α)Proof. The identity |~v× ~w|2= |~v|2| ~w|2−(~v· ~w)2can be proven by direct computation. Now, |~v· ~w| = |~v|| ~w| cos(α).AREA. The length |~v × ~w| is the area of the parallelogram spanned by ~v and ~w.Proof. Because | ~w| sin(α) is the height of the parallelogram with base length |~v|, the area is |~v|| ~w| sin(α) whichis by the above formula equal to |~v × ~w|.EXAMPLE. If ~v = (a, 0, 0) and ~w = (b cos(α), b sin(α), 0), then ~v × ~w = (0, 0, ab sin(α)) which has length|ab sin(α)|.ZERO CROSS PRODUCT. We see that ~v × ~w is zero if ~v and ~w are parallel.ORIENTATION. The vectors ~v, ~w and ~v × ~w form a right handedcoordinate system. The right hand rule is: put the first vector ~v onthe thumb, the second vector ~w on the pointing finger and the thirdvector ~v × ~w on the third middle finger.EXAMPLE.~i,~j,~i ×~j =~k forms a right handed coordinate system.DOT PRODUCT (is a scalar)~v · ~w = ~w · ~v commutative|~v · ~w| = |~v|| ~w| cos(α) angle(a~v) · ~w = a(~v · ~w) linearity(~u + ~v) · ~w = ~u · ~w + ~v · ~w distributivity{1, 2, 3}.{3, 4, 5} in Mathematicaddt(~v · ~w) = ˙v · ~w + ~v · ˙w product ruleCROSS PRODUCT (is a vector)~v × ~w = − ~w × ~v anti-commutative|~v × ~w| = |~v|| ~w| sin(α) angle(a~v) × ~w = a(~v × ~w) linearity(~u + ~v) × ~w = ~u × ~w + ~v × ~w distributivityCross[{1, 2, 3}, {3, 4, 5}] in Mathematicaddt(~v × ~w) =˙~v × ~w + ~v ×˙~w product ruleTRIPLE SCALAR PRODUCT. The scalar [~u,~v, ~w] = ~u · (~v × ~w) is called the triple scalar product of ~u, ~v, ~w.PARALLELEPIPED. [~u, ~v, ~w] is the volume of the parallelepipedspanned by ~u, ~v, ~w because h = ~u · ~n/|~n| is the height of the par-allelepiped if ~n = (~v × ~w) is a normal vector to the ground par-allelogram which has area A = |~n| = |~v × ~w|. The volume of theparallelepiped is hA = ~u · ~n|~n|/|~n| = |~u · (~v × ~w)|.EXAMPLE. Find the volume of the parallel epiped which has the one corner O = (1, 1, 0) and three cornersP = (2, 3, 1), Q = (4, 3, 1), R = (1, 4, 1) connected to it.ANSWER: The parallelepiped is spanned by ~u = (1, 2, 1), ~v = (3, 2, 1), and ~w = (0, 3, 2). We get ~v × ~w =(1, −6, 9) and ~u · (~v × ~w) = −2. The volume is 2.DISTANCE POINT-LINE (3D). If P is a point in space and L is the line whichcontains the vector ~u, thend(P, L) = |~P Q × ~u|/|~u|is the distance between P and the line L.PLANE THROUGH 3 POINTS P, Q, R:The vector ~n =~P Q ×~P R is orthogonal to the plane. We will next week that ~n = (a, b, c) defines the planeax + by + cz = d, with d = ax0+ by0+ cz0which passes through the points P = (x0, y0, z0), Q, R.The cross product appears in many different applications:ANGULAR MOMENTUM. If a mass point of mass m moves along a curve ~r(t), then the vector~L(t) =m~r(t) × ~r0(t) is called the angular momentum of the point. It is coordinate system dependent.ANGULAR MOMENTUM CONSERVATION.ddt~L(t) = m~r0(t) × ~r0(t) + m~r(t) × ~r00(t) = ~r(t) ×~F (t)In a central field, where~F (t) is parallel to ~r(t), we get d/dtL(t) = 0 which means L(t) is constant.TORQUE. In physics, the quantity ~r(t)×~F (t) is also called the torque. The time derivative of the momentumm~r0is the force, the time derivative of the angular momentum~L is the torque.KEPLER’S AREA LAW. (Proof by Newton)The fact that~L(t) is constant means first of allthat ~r(t) stays in a plane spanned by ~r(0) and~r0(0). The experimental fact that the vector ~r(t)sweeps over equal areas in equal times ex-presses angular momentum conservation: |~r(t) ×~r0(t)dt/2| = |~Ldt/m/2| is the area of a smalltriangle. The vector ~r(t) sweeps over an areaRT0|~L|dt/(2m) = |~L|T/(2m) in time [0, T ].r(t)SunEarthr’(t)dr x r/2MORE PLACES IN PHYSICS WHERE THE CROSS PRODUCT OCCURS:The top, the motion of a rigid body is describe by the angular momentum L and the angular velocity vectorΩ in the body. Then˙L = L × Ω + M , where M is an external torque obtained by external forces.Electromagnetism: (informal) a particle moving along ~r(t) in a magnetic field~B for example experiences the force~F (t) = q~r0(t) ×~B, where q is the charge ofthe particle. In a constant magnetic field, the particles move on circles: if m isthe mass of the particle, then m~r00(t) = q~r0(t) ×~B implies m~r0(t) = q~r (t) ×~B.Now d/dt|~r|2= 2~r · ~r0= ~r · q~r (t) ×~B = 0 so that |~r| is constant.Hurricanes are powerful storms with wind velocities of 74 miles per hour or more.On the northern hemisphere, hurricanes turn counterclockwise, on the southernhemisphere clockwise. This is a feature of all low pressure systems and can beexplained by the Coriolis force. In a rotating coordinate system a particle of massm moving along ~r(t) experience the following forces: m~ω0×~r (inertia of rotation),2m~ω ×~r0(Coriolis force) and mω × (~ω × ~r)) (Centrifugal force). The Coriolis forceis also responsible for the circulation in Jupiter’s Red
View Full Document
Unlocking...