MA1511 Chapter 3 Lecture Slides.pdf - Chapter 3...

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1 Motion in space 𝐫𝐫 ( 𝑡𝑡 ) = 𝑓𝑓 ( 𝑡𝑡 ) 𝑔𝑔 ( 𝑡𝑡 ) ( 𝑡𝑡 ) (displacement) 𝐫𝐫 𝑡𝑡 = 𝑑𝑑 𝑑𝑑𝑡𝑡 𝐫𝐫 𝑡𝑡 = 𝑓𝑓 𝑡𝑡 𝑔𝑔 𝑡𝑡 𝑡𝑡 velocity 𝐫𝐫 ′′ 𝑡𝑡 = 𝑑𝑑 2 𝑑𝑑𝑡𝑡 2 𝐫𝐫 𝑡𝑡 = 𝑓𝑓 ′′ 𝑡𝑡 𝑔𝑔 ′′ 𝑡𝑡 ′′ 𝑡𝑡 (acceleration) 𝐫𝐫 𝑡𝑡 is tangential to the curve Chapter 3 Vector–Valued Functions 2-D Case Simply remove the third component 3-D Case 𝐫𝐫 ( 𝑡𝑡 ) = 𝑓𝑓 ( 𝑡𝑡 ) 𝑔𝑔 ( 𝑡𝑡 ) 𝐫𝐫 𝑡𝑡 = 𝑓𝑓𝑓 ( 𝑡𝑡 ) 𝑔𝑔𝑓 ( 𝑡𝑡 ) and so on Dr NG Wee Seng [email protected] Cartesian to Parametric 𝑦𝑦 = 𝑓𝑓 ( 𝑥𝑥 ) can be represented by the vector function 𝐫𝐫 ( 𝑡𝑡 ) =
2 Motion in space Chapter 3 Vector–Valued Functions Crossing of Paths & Collision Given two space curves 𝐫𝐫 𝟏𝟏 ( 𝑡𝑡 ) & 𝐫𝐫 𝟐𝟐 ( 𝑡𝑡 ) traced by two moving particles, Q1 will their paths cross ? Method: Solve 𝐫𝐫 𝟏𝟏 ( 𝑢𝑢 ) = 𝐫𝐫 𝟐𝟐 ( 𝑣𝑣 ) for some 𝒖𝒖 , 𝒗𝒗 Q2 will they collide ? 𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌 : 𝐒𝐒𝐌𝐌𝐒𝐒𝐒𝐒𝐌𝐌 𝐫𝐫 𝟏𝟏 ( 𝑡𝑡 ) = 𝐫𝐫 𝟐𝟐 ( 𝑡𝑡 ) for some 𝒕𝒕 A simple linear example 𝐫𝐫 𝟏𝟏 ( 𝑡𝑡 ) = 𝑡𝑡 2 2𝑡𝑡 + 1 𝐫𝐫 𝟐𝟐 ( 𝑡𝑡 ) = 4 𝑡𝑡 4 𝑡𝑡 + 1 Dr NG Wee Seng [email protected] A non-linear Example (Ans: paths cross at (4, 5); no collision )
3 Tangent Lines Equation of Tangent Line Example We are given the space curve C : The tangent to the curve at a point P is perpendicular to −𝐢𝐢 + 𝐣𝐣 + 𝐤𝐤 Find an equation of the tangent at P.