![]() ![]() ![]() (~r − ~r0) = 0 where ~n is the vector orthogonal to every vector in the given plane and ~r − ~r0 is the vector between any two points on the plane.The symmetric equations for a line passing through the points (x0, y0, z0) and (x1, y1, z1) are given by: x− x0 x1 − x0 = y − y0 y1 − y0 = z − z0 z1 − z0 6 9.13 Segment of a Line The line segment from ~r0 to ~r1 is given by: ~r(t) = (1− t)~r0 + t~r1 for 0 ≤ t ≤ 1 9.14 Vector Equation of a Plane ~n (~a×~b) 9.11 Vector Equation of a Line ~r = ~r0 + t~v 9.12 Symmetric Equations of a Line x− x0 a = y − y0 b = z − z0 c where the vector ~c = 〈a, b, c〉 is the direction of the line.~a×~b = −~b× ~a (c~a)×~b = c(~a×~b) = ~a× (c~b) ~a× (~b+ ~c) = ~a×~b+ ~a× ~c (~a+~b)× ~c = ~a× ~c+~b× ~c 9.10 Scalar Triple Product The volume of the parallelpiped determined by vectors ~a, ~b, and ~c is the magnitude of their scalar triple product: V = |~a ~a×~b = 〈a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1〉 9.9 Properties of the Cross Product Two vectors are parallel if their cross product is 0. ~b |~a| ) ~a |~a| 9.8 Cross Product ~a×~b = (|~a||~b| sin θ)~n where ~n is the unit vector orthogonal to both ~a and ~b.~b |~a| Vector projection of ~b onto ~a: proj~a ~b = ( ~a.~a = 0 5 9.7 Vector Projections Scalar projection of ~b onto ~a: comp~a ~b = ~a.~b = a1b1 + a2b2 + a3b3 9.6 Properties of the Dot Product Two vectors are orthogonal if their dot product is 0.The unit vector ~u in the same direction as ~a is given by: ~u = ~a |~a| 9.5 Dot Product ~a 10 2 9 Vectors and the Geometry of Space 9.1 Distance Formula in 3 Dimensions The distance between the points P1(x1, y1, z1) and P2(x2, y2, z2) is given by: |P1P2| = √ (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2 9.2 Equation of a Sphere The equation of a sphere with center (h, k, l) and radius r is given by: (x− h)2 + (y − k)2 + (z − l)2 = r2 9.3 Properties of Vectors If ~a,~b, and ~c are vectors and c and d are scalars: ~a+~b = ~b+ ~a ~a+ 0 = ~a ~a+ (~b+ ~c) = (~a+~b) + ~c ~a+−~a = 0 c(~a+~b) = c~a+ c+~b (c+ d)~a = c~a+ d~a (cd)~a = c(d~a) 9.4 Unit Vector A unit vector is a vector whose length is 1. 10 10.12Equations of a Parametric Surface. ![]() 10 10.11Tangential and Normal Components of Acceleration. 10 10.10Parametric Equations of Trajectory. 9 10.4 Derivative Rules for Vector Functions. 8 10 Vector Functions 9 10.1 Limit of a Vector Function. Volume of a parallelepiped v) Vector equation of a line r r0 tv A parametric form of the equation of a line x x0 tv1, y y0 tv2, z z0 tv3 Vector normal form of the equation of a plane n ( r r0 ) 0 equation of a plane A ( x x0 ) B ( y y0 ) C ( z z0 ) 0 v v Functions and Motion in Space Velocity of a particle dr, where r(t) x (t)i y(t)j z(t)k is the position dt Acceleration of a particle dv, where v is the velocity dt Arc Length of a smooth curve Rb L a where r(t) x (t)i y(t)j z(t)k is traced exactly once as t increase on the interval a, Curvature of a smooth curve 1 dT 0 dt where T r0 is the unit tangent vector.Download Calculus 3 formula sheet and more Calculus Cheat Sheet in PDF only on Docsity!Harvard College Math 21a: Multivariable Calculus Formula and Theorem Review 1 Contents Table of Contents 4 9 Vectors and the Geometry of Space 5 9.1 Distance Formula in 3 Dimensions. Preview text Vectors and the Geometry of Space q Magnitude of a vector v21 v22 v23 where v hv1, v2, v3 i Dot product u v u1 v1 u2 v2 u3 v3 where u hu1, u2, u3 i and v hv1, v2, v3 i u v cos where is the angle between u and v Vector projection of u onto v projv u Cross product u v h u2 v3 u3 v2, u3 v1 u1 v3, u1 v2 u2 v1 i where u hu1, u2, u3 i and v hv1, v2, v3 i sin where is the angle between u and v Area of a parallelogram where u and v form two sides of the parallelogram. ANSC511 Lab2 Skeletal System Objective Sheet KEY. ![]()
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