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6. (a) Let I be an interval and ~r₁(t) = (x₁(t),y₁(t),z₁(t)) where t = I, be two differentiable curves in R³ Show that i. and ~r₂(t) = (x₂(t),y₂(t),z2(t)), ii. d (F₁(t) · F₂(t)) = F'₁(t) • F₂(t) + r1(t) · Tº'₂(t) dt d ; (Fi(t) × F₂(t)) = r₁(t) × r₂(t) + ₁(t) × T₂(t) dt

6. (a) Let I be an interval and ~r₁(t) = (x₁(t),y₁(t),z₁(t)) where t = I, be two differentiable curves in R³ Show that i. and ~r₂(t) = (x₂(t),y₂(t),z2(t)), ii. d (F₁(t) · F₂(t)) = F'₁(t) • F₂(t) + r1(t) · Tº'₂(t) dt d ; (Fi(t) × F₂(t)) = r₁(t) × r₂(t) + ₁(t) × T₂(t) dt

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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6. (a) Let I be an interval and ~r₁(t) = (x₁(t),y₁(t),z₁(t)) where t€ 1, be two differentiable curves in R³ Show that i. ii. and ~r₂(t) = (x₂(t),y₂(t),z2(t)), d (F₁(t) · F₂(t)) = ri(t) · F₂(t) + Fi(t) · F₂(t) dt d dt (b) * Suppose that I is an interval and (Fi(t) × F₂(t)) = T₁(t) × ²₂2(t) + r₁(t) × F2(t) X ~r(t) = (x(t),y(t),z(t)), where t € I, is a twice-differentiable curve that describes the position of an object in R³. If the object is moving at a constant speed, show that its velocity is always perpendicular to its acceleration.
Transcribed Image Text:6. (a) Let I be an interval and ~r₁(t) = (x₁(t),y₁(t),z₁(t)) where t€ 1, be two differentiable curves in R³ Show that i. ii. and ~r₂(t) = (x₂(t),y₂(t),z2(t)), d (F₁(t) · F₂(t)) = ri(t) · F₂(t) + Fi(t) · F₂(t) dt d dt (b) * Suppose that I is an interval and (Fi(t) × F₂(t)) = T₁(t) × ²₂2(t) + r₁(t) × F2(t) X ~r(t) = (x(t),y(t),z(t)), where t € I, is a twice-differentiable curve that describes the position of an object in R³. If the object is moving at a constant speed, show that its velocity is always perpendicular to its acceleration.
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