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a) Let the 3-D vector a make angles a, B and y with the coordinate axes. Show that = cos ai + cos Bj + cos yk is a unit vector in the direction of the vector a. Show also that cos?a + cos?B + cos?y = 1. %3D

a) Let the 3-D vector a make angles a, B and y with the coordinate axes. Show that = cos ai + cos Bj + cos yk is a unit vector in the direction of the vector a. Show also that cos?a + cos?B + cos?y = 1. %3D

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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a) Let the 3-D vector a make angles a, ß and y with the coordinate axes. Show that â = cos ai + cos Bj + cos yk is a unit vector in the direction of the vector a. Show also that cos²a + cos?B + cos²y = 1. b) The path of a particle is given by the position vector 7(t) = a sin ti + a cos tj + b tk, where a and b are real constants. i) Sketch the graph of the curve traced by the particle as t varies. ii) Find the length of the curve between the points correspondong to t = 0 and t = 27. c) The displacement of a particle in space is given by the vector r'(t) = sint cost i+sin²t }+cost k. Find the tangent, the normal and binormal, and the curvature and torsion of the curve at the points (i) t= 0 and (ii) t = "/2.
Transcribed Image Text:a) Let the 3-D vector a make angles a, ß and y with the coordinate axes. Show that â = cos ai + cos Bj + cos yk is a unit vector in the direction of the vector a. Show also that cos²a + cos?B + cos²y = 1. b) The path of a particle is given by the position vector 7(t) = a sin ti + a cos tj + b tk, where a and b are real constants. i) Sketch the graph of the curve traced by the particle as t varies. ii) Find the length of the curve between the points correspondong to t = 0 and t = 27. c) The displacement of a particle in space is given by the vector r'(t) = sint cost i+sin²t }+cost k. Find the tangent, the normal and binormal, and the curvature and torsion of the curve at the points (i) t= 0 and (ii) t = "/2.
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