9/28/2002, CROSS PRODUCT Math 21a, O. KnillCROSS PRODUCT. The cross product of two vectors v = (v1, v2, v3) and w = (w1, w2, w3) is defined asv × w = (v2w3− v3w2, v3w1− v1w3, v1w2− v2w1).AREA. v × w is orthogonal to v and orthogonal to w. Its length |v × w| is the area of parallelogram spannedby v and w. Proof. Check it first for v = (1, 0, 0) and w = (cos(α), sin(α), 0), where v × w = (0, 0, sin(α)) haslength | sin(α)| which is indeed the area of the parallelogram spanned by v and w. A more general case can beobtained by scaling v and w: both the area as well as the cross product behave linearly in v and w.The formula|v × w| = |v||w| sin(α)(which can be checked also using |v × w|2= |v|2|w|2− (v · w)2and |v · w| = |v||w| cos(α), gives an other way tomeasure angles. We see that v × w is zero if v and w are parallel or one of the vectors is zero.DOT PRODUCT (is scalar)v · w = w · v commutative|v · w| = |v||w| cos(α) angle(av) · 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 vector)v × w = −w × v anti-commutative|v × w| = |v||w| sin(α) angle(av) × w = a(v × w) linearity(u + v) × w = u × w + v × w distributivityCross[{1, 2, 3}, {3, 4, 5}] Mathematicaddt(v × w) = ˙v × w + v × ˙w product ruleTRIPLE SCALAR PRODUCT. The scalars [u, v, w] = u · v × w is called the triple scalar product of u, v, w.It is the volume of the parallelepiped spanned by u, v, w because u · n is the height of the parallelepiped if n isa normal vector to the ground parallelogram which has area |v × w|.DISTANCE POINT-PLANE (3D). If P is a point in space and n · x = d is aplane containing a point Q, thend(P, L) = |(P − Q) · n|/|n|is the distance between P and the plane.DISTANCE POINT-LINE (3D). If P is a point in space and L is the liner(t) = Q + tu, thend(P, L) = |(P − Q) × u|/|u|is the distance between P and the line L.DISTANCE LINE-LINE (3D). L is the line r(t) = Q + tu and M is the lines(t) = P + tv, thend(L, M) = |(P − Q) · (u × v)|/|u × v|is the distance between the two lines L and M .PLANE THROUGH 3 POINTS P, Q, R: The vector (a, b, c) = n = (Q − P ) × (R − P ) is normal to the plane.Therefore, the equation is ax + by + cz = d. The constant is d = ax0+ by0+ cz0because P = (x0, y0, z0) mustbe on the plane.PLANE THROUGH POINT P AND LINE r(t) = Q + tu. The vector (a, b, c) = n = u × (Q − P ) is normal tothe plane. Therefore the plane is given by ax + by + cz = d, where d = ax0+ by0+ cz0and P = (x0, y0, z0).LINE ORTHOGONAL TO PLANE ax+by+cz=d THROUGH POINT P. The vector n = (a, b, c) is normal tothe plane. The line is r(t) = P + nt.ANGLE BETWEEN PLANES. The angle between the two planes a1x+b1y + c1z = d1and a2x+b2y + c2z = d2is arccos(n1·n2|n1||n2|, where ni= (ai, bi, ci). Alternatively, it is arcsin(n1×n2|n1||n2|.INTERSECTION BETWEEN TWO PLANES. Find the line which is the intersection of two non-parallelplanes a1x + b1y + c1z = d1and a2x + b2y + c2z = d2. Find first a point P which is in the intersection. Thenr(t) = P + t(n1× n2) is the line, we were looking for.ANGULAR MOMENTUM. If a mass point of mass m moves along a curve r(t), then the vector L(t) =mr(t) × r0(t) is called the angular momentum.ANGULAR MOMENTUM CONSERVATION.ddtL(t) = mr0(t) × r0(t) + mr(t) × r00(t) = r(t) × F (t)In a central field, where F (t) is parallel to r(t), this vanishes.TORQUE. The quantity r(t) × F (t) is called the torque. The time derivative of the momentum mr0is theforce, 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) andr0(0). The experimental fact that the vectorr(t) sweeps over equal areas in equal timesexpresses the angular momentum conservation:|r(t) × r0(t)dt/2| = |Ldt/m/2| is the area of asmall triangle. The vector r(t) sweeps over anareaRT0Ldt/(2m) = LT/(2m) in time [0, T ].r(t)SunEarthr’(t)dr x r/2PLACES IN PHYSICS WHERE THE CROSS PRODUCT OCCURS: (informal)In a rotating coordinate system a particle of mass m moving along r(t) experience the following forces: mω0× r(inertia of rotation), 2mω × r0(Coriolis force) and mω × (ω × r)) (Centrifugal force).The top, the motion of a rigid body is describe by the angular momentum M and the angular velocity vectorΩ in the body. Then˙M = M × Ω + F , where F is an external force.Electromagnetism: a particle moving along r(t) in a magnetic field B for example experiences the forceF (t) = qr0(t) × B, where q is the charge of the
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