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Math 103 – Fall 2007 Chapter 12 Notes Alex Beutel September 16, 2007 Defintion: A vector v in the Cartesian plane is an orderedpair of real numbers that has the form < a, b >. We write v=< a, b > and call a and b the components of vector v. a • b =| a || b | cosΘ Direction Angles a1 a•i = cosα = |a| |i| |a| a•j a2 cosβ = = |a| |j| |a| a•k a3 cosγ = = |a| |k| |a| cos2 α + cos2 β + cos2 γ = 1 a•b | a || b | cosΘ = b |b| compb a =| a | cosΘ compb a = Then manipula formula 1 to get rid of cosΘ REVIEW 12.2 - DIRECTIONS, ANGLES, AND PROJECTIONS To prove perpendicularity of cross product show that dot product is 0. | a x b | =| a || b | sinΘ a1 a • (b x c) = b1 c1 a2 b2 c2 a3 b3 c3 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) V =| a • (b x c) | That is the volume of a parallelepiped determined by vectors a, b, and c. The volume of a pyramid is 1 that of a parallelepiped. 6 τ =rxF F = (qv) x B END OF 12.3 12.4 (13) (14) 1 Symmetric Equtions: y − y0 z − z0 x − x0 = = a b c Plane in Space: n = a, b, c P0 = (x0 , y0 , z0 ) a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0 ax + by + cz = d (15) (16) (17) (18) (19) (20) To measure the angle of intersection between two planes measure the angle between the normal vecotrs (ie n and m) n•m | n || m | cosΘ = 12.5 seems pointless???? 12.6: s = arc length b b (21) (22) b s= a | v(t) | dt = a v(t)dt = a dx dt 2 + dy dt 2 + dz dt 2 (23) (24) (25) (26) ds =v dt v(t) T(t) = v(t) T is the unit tangent vector meaning its tangent to the curve but has a length of 1. |xy −x y | [(x )2 + (y 3 3 )2 ] 2 κ= κ= = |xy −x y | v3 | d2 y dx2 | 3 dy ( dx )2 ] 2 (27) (28) (29) (30) (31) |y | [1 + (y )2 ] 2 = [1 + dT = κN ds N points in the direction that the curve is bending??? See page 820 for better explanation: Draw a circle (known as the osculating circle) which is tangent to T. The radius is ρ and points in the direction of N. 1 ρ= κ dv v•a r (t) • r (t) = = dt v | r (t) | 2 v | r (t) x r (t) | = aN = κv 2 = ρ | r (t) | |vxa| | r (t) x r (t) | κ= = v3 | r (t) |3 aT = (32) (33) (34) 2 

Document Info

Posted By:

Alex Beutel

Date:

Thursday, December 27, 2007

School:

Duke University

Class:

Math 103

Tags:

• Study Guide
• Formula Sheet
• Curvature
• Vectors
• Calc3
• Calculus
• Math

About this Document

This is a more rough study guide from our first exam for Math 103 (Multivariable Calculus). This includes vectors, dot products, cross products, other vector properties, arc length, curvature (kappa), among other topics. Not as clean but still useful.