Classical Dynamics of Particles and Systems
5th Edition
ISBN: 9780534408961
Author: Stephen T. Thornton, Jerry B. Marion
Publisher: Cengage Learning
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Chapter 1, Problem 1.11P
To determine
Show that the triple scalar product can be written as
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Given two vectors A⃗ = 4.20 i^+ 7.60 j^ and B⃗ = 5.70 i^− 2.40 j^ , find the scalar product of the two vectors A⃗ and B⃗ .
Find the angle between these two vectors.
For the pair of vectors A = (6.00î + 4.00j) and B = (9.00î – 6.00j) in the xy plane, determine the following. (Enter all angle answers between 0 and 180°.)
(a) The scalar product
A·B =
(b) The angle 0 between the vectors
(c) The angles a and B which are respectively the (smallest) angles between the vector A and the positive x and positive y axes
a =
B =
(d) The angles y and 8, which are respectively the (smallest) angles between the vector B and the positive x and positive y axes
8 =
Use the definition of scalar product, á · b = ab cos 0, and the fact that á ·
= axbx + ayby + azbz to calculate the angle between the two vectors given by a = 2.0i + 2.0j + 2.0k and b = 9.0î + 7.0î + 3.0k.
Chapter 1 Solutions
Classical Dynamics of Particles and Systems
Ch. 1 - Prob. 1.1PCh. 1 - 1.2. Prove Equations 1.10 and 1.11 from...Ch. 1 - Prob. 1.3PCh. 1 - Show
(a) (AB)t = BtAt (b) (AB)−1 = B−1 A−1
Ch. 1 - Prob. 1.5PCh. 1 - Prob. 1.6PCh. 1 - Consider a unit cube with one corner at the origin...Ch. 1 - Prob. 1.8PCh. 1 - For the two vectors
find
A − B and |A –...Ch. 1 - A particle moves in a plane elliptical orbit...
Ch. 1 - Prob. 1.11PCh. 1 - Let a, b, c be three constant vectors drawn from...Ch. 1 - X is an unknown vector satisfying the following...Ch. 1 - Prob. 1.14PCh. 1 - Prob. 1.15PCh. 1 - What surface is represented by r a = const, that...Ch. 1 - Obtain the cosine law of plane trigonometry by...Ch. 1 - Obtain the sine law of plane trigonometry by...Ch. 1 - Prob. 1.19PCh. 1 - 1-20. Show that
Ch. 1 - Show (see also Problem 1–11) that
Ch. 1 - Prob. 1.22PCh. 1 - Use the εijk notation and derive the identity
(A ×...Ch. 1 - Prob. 1.24PCh. 1 - Find the components of the acceleration vector a...Ch. 1 - Prob. 1.26PCh. 1 - If r and are both explicit functions of time,...Ch. 1 - Show that
Ch. 1 - Prob. 1.29PCh. 1 - Prob. 1.30PCh. 1 - Show that
(a)
(b)
(c)
Ch. 1 - Show that (2arr+2brr)dt=ar2+br2+const. where r is...Ch. 1 - Show that (rrrrr2)dt=rr+C where C is a constant...Ch. 1 - Prob. 1.34PCh. 1 - Prob. 1.35PCh. 1 - Prob. 1.36PCh. 1 - Prob. 1.37PCh. 1 - Prob. 1.38PCh. 1 - A plane passes through the three points (x, y, z)...Ch. 1 - For what values of a are the vectors A = 2ai − 2j...
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Similar questions
- Find the angle between vectors for (a) D=(-3.0i-4.0j)m and A=(-3.0i+4.0j)m and (b) D=(2.0i+4.0j+K)m and B=(-2.0i+3.0j+2.0K)m .arrow_forwardWhich of the following statements is true about the relationship between the dot product of two vectors and the product of the magnitudes of the vectors? (a) AB is larger than AB. (b) AB is smaller than AB. (c) AB could be larger or smaller than AB, depending on the angle between the vectors. (d) AB could be equal to AB.arrow_forwardFor the two vectors find A − B and |A – B| component of B along A angle between A and B A × B (A − B) × (A + B)arrow_forward
- Find the scalar components of three-dimensional vectors G and H in the following figure and write the vectors in vector component form in terms of the unit vectors of the axes.arrow_forwardFor the pair of vectors A = (6.00î + 4.00ĵ) and B = (9.0oî – 6.00ĵ) in the xy plane, determine the following. (Enter all angle answers between 0 and 180°.) %3D (a) The scalar product A: B = (b) The angle 0 between the vectors (c) The angles a and ß which are respectively the (smallest) angles between the vector A and the positive x and positive y axes a = %D (d) The angles y and 8, which are respectively the (smallest) angles between the vector B and the positive x and positive y axes Y 8 %3Darrow_forwardUse the definition of scalar product, ?→⋅?→�→⋅�→ = ab cos θ, and the fact that ?→⋅?→�→⋅�→ = axbx + ayby + azbz to calculate the angle between the two vectors given by ?→=6.0?̂ +6.0?̂+6.0?̂�→=6.0�̂+6.0�̂+6.0�̂ and ?→=2.0?̂ +2.0?̂+3.0?̂�→=2.0�̂+2.0�̂+3.0�̂.arrow_forward
- Using the definition of the scalar product, find the angle between the following vectors. (Find the smallest nonnegative angle.) (a) A = 4î − 5ĵ and B = 7î − 5ĵ (b) A = −8î + 5ĵ and B = 3î − 4ĵ + 2 (c) A = î −2ĵ + 2 and B = 3ĵ + 4arrow_forwardGiven two vectors A=4.00i + 7.00j and B=5.00i - 2.00j. Find the scalar product and vector product of the two vectors.arrow_forwardUsing the definition of the scalar product, find the angle between the following vectors. (Find the smallest nonnegative angle.) (a) A = 2î − 9ĵ and B = 5î − 6ĵ °(b) A = −3î + 3ĵ and B = 3î − 4ĵ + 2 °(c) A = î −2ĵ + 2 and B = 3ĵ + 4 °arrow_forward
- Problem 5: Two vectors in Cartesian coordinates have components A = (3, 1) and B = (2, y). a) If the scalar product of A and B is 8, what is y?arrow_forwardUsing the definition of dot product: Ax B= (A,B₂ - A₂By)i + (A₂Bx − AxB₂)j + (AxBy - Ay Bx)k Find the cross product Ax B the following vectors: (a) U A = 11 + 3) + 0k B = 3î + 1ĵ + 0k A = 11 - 3j+2k B = -31 + 1ĵ+ 0k (b) (d) A = 1î + 1ĵ + Ok B = 2î - 3j+0k A = 21-5j-1k B = 3î + 1ĵ+ 3karrow_forwardVector A has a magnitude of 4.0 at 530 counterclockwise from + x-axis and Vector B has a magnitude of 5.0 at 1300 from + x-axis. Find the scalar product of A∙Barrow_forward
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