The document contains 9 multiple choice questions regarding vectors and coordinate systems. It provides the questions, correct answers, and explanations of the reasoning. The questions cover topics like the dot product of vectors, finding the angle between vectors, Cartesian and polar coordinate systems, and representing points on spheres and cylinders using spherical and cylindrical coordinates.
The document contains 9 multiple choice questions regarding vectors and coordinate systems. It provides the questions, correct answers, and explanations of the reasoning. The questions cover topics like the dot product of vectors, finding the angle between vectors, Cartesian and polar coordinate systems, and representing points on spheres and cylinders using spherical and cylindrical coordinates.
The document contains 9 multiple choice questions regarding vectors and coordinate systems. It provides the questions, correct answers, and explanations of the reasoning. The questions cover topics like the dot product of vectors, finding the angle between vectors, Cartesian and polar coordinate systems, and representing points on spheres and cylinders using spherical and cylindrical coordinates.
The document contains 9 multiple choice questions regarding vectors and coordinate systems. It provides the questions, correct answers, and explanations of the reasoning. The questions cover topics like the dot product of vectors, finding the angle between vectors, Cartesian and polar coordinate systems, and representing points on spheres and cylinders using spherical and cylindrical coordinates.
Multiple-choice questions may continue on the next column or page find all choices before answering. 001 (part 1 of 3) 10.0 points The dot or scalar product of two (3d) vectors ~a = ha1 , a2, a3 i and ~b = hb1 , b2, b3 i is defined as ~a ~b =
3 X
a i bi = a 1 b1 + a 2 b2 + a 3 b3 .
i=1
Explanation: From the definition, this is simply calculated by ~a ~b = a1 b1 + a2 b2 + a3 b3 = 10 3.3 + 7.5 6.75 + 0 0 = 83.625 . 002 (part 2 of 3) 10.0 points Determine , the angle between ~a and ~b. Express your answer in degrees. Correct answer: 27.0766. Explanation: From the geometric interpretation of the scalar product, ! ~b ~ a
= cos1 k~a kk~b k
In 3d Euclidean space, the scalar product
has a geometric interpretation, given by ~a ~b =k~a kk~b k cos ,
~ ~a b
q = cos1 q 2 2 2 2 2 2 a 1 + a 2 + a 3 b1 + b2 + b3
83.625 1 = cos 12.5 7.51349 = 27.0766 . 003 (part 3 of 3) 10.0 points
where is the angle between ~a and ~b and the
norm(length) of a vector k~a k is defined by
k~a k= ~a ~a .
If two vectors are perpendicular, what is their
In words, the scalar product of two vectors can
be thought of as the product of the magnitude of ~a with the magnitude of the projection of ~b onto the direction of ~a. It is used to calculate the product of vector quantities when only the parallel components of each vector contribute (e.g., Work = Force Displacement).
Correct answer: 0.
Let ~a = h10, 7.5, 0i and ~b = h3.3, 6.75, 0i.
Calculate ~a ~b.
Correct answer: 83.625.
scalar product?
Explanation: From the geometric interpretation, one can see that when = 90 , the scalar product is equal to zero. 004 (part 1 of 3) 10.0 points Recall that, in a 2d plane polar coordinate system, a point in space is described by its radial distance r from the origin and the angle
sheriff (aas3563) HW1-a li (55140)
between the +x-axis and a line drawn from
the origin to the point. For example, the point (3, /4) looks like:
The relations between cartesian (x, y) coordinates and plane polar (r, ) are given by x = r cos , y = r sin , x2 + y 2 = r 2 . Question: A circle of radius R centered at the origin is described in plane polar coordinates by r = R. What are the Cartesian (i.e., (x,y)) coordinates of a point on the circle at some arbitrary angle , expressed in terms of polar quantities? (Note: you will frequently parametrize a point on a circle in this way.) 1. (R, ) 2. (R sin , R cos )
2 2 1 y 3. (x + y , tan ) x 4. (x, y) 5. (x cos , y sin ) 6. (R cos , R sin ) correct Explanation: From the Cartesian/plane polar relations given above, the x-coordinate is R cos and the y-coordinate is R sin , so (R cos , R sin ) is the correct choice. 005 (part 2 of 3) 10.0 points In 3d, we may extend this idea to cylindrical coordinates (r, , z), which you may think of as a plane polar coordinate system with the z-dimension tacked on.
This coordinate system is useful when dealing with problems that have radial symmetry about some central axis. Question: You are given a hollow cylinder of radius R whose central axis is the z-axis and whose base rests on the xy-plane. What are the Cartesian coordinates of an arbitrary point on the surface of the cylinder in terms of cylindrical quantities?
2 2 1 y 1. (x + y , tan , z) x 2. (R sin , R cos , z) 3. (R cos , R sin , z) correct 4. (R, , z) 5. (x cos , y sin , z) 6. (x, y, z) Explanation: Since the cylindrical coordinate system is just a plane polar system with the z-axis appended to it, the correct answer is the same as in 1 above, but with the z coordinate added: (R cos , R sin , z) 006 (part 3 of 3) 10.0 points In cases where we have radial symmetry about the origin (usually called spherical symmetry), it is most useful to use spherical coordinates (r, , ). In the spherical coordinate system, a point is defined by: its radial distance from the origin r; the azimuthal angle , defined as the angle between the +x-axis and the projection on the xy-plane of the line
sheriff (aas3563) HW1-a li (55140)
drawn from the origin to the point; the polar angle , defined as the angle between the +z axis and the line drawn from the origin to the point. Note that for the polar angle, 0 .
A particle undergoes two displacements.
The first has a magnitude of 189 cm and makes an angle of 110 with the positive x axis. The resultant displacement has a magnitude of 155 cm and is directed at an angle of 31.6 to the positive x axis. Find the magnitude of the second displacement. Correct answer: 219.009 cm. Explanation: Let :
R1 = 189 cm R = 155 cm
Question: You are given a solid sphere of
radius R centered at the origin. What are the Cartesian coordinates of an arbitrary point within the sphere in terms of spherical quantities. You will need to use the figure to determine the x,y, and z coordinates in terms of their spherical counterparts.
r1
3. (r sin cos , r sin sin , r cos ) correct
4. (r sin cos , r sin sin , r cos ) 5. (R sin cos , R sin sin , R cos ) 6. (r cos cos , r cos sin , r sin ) Explanation: Examining the figure, the projection onto the xy plane of the line drawn from the origin to an arbitrary point is seen to be r sin . The component of this projection along the x-axis is r sin cos , and the projection along the y-axis is r sin sin . Finally, the projection onto the z-axis of the line drawn from the origin to the arbirary point is seen to be r cos . Combining everything, the correct answer is (r sin cos , r sin sin , r cos ). 007 (part 1 of 2) 10.0 points
1 = 110 and = 31.6 .
r2
R = R1 + R2 R2 = R R1 ,
1. (x, y, z) 2. (r sin sin , r sin cos , r cos )
at at
so
R2x = (155 cm) cos 31.6 (189 cm) cos 110
= 196.659 cm , and R2y = (155 cm) sin 31.6 (189 cm) sin 110 = 96.3841 cm , and the magnitude is q 2 + R2 R2x 2y q = (196.659 cm)2 + (96.3841 cm)2
|R2 | =
= 219.009 cm .
008 (part 2 of 2) 10.0 points
What is the direction of the second displacement (with positive measured counterclockwise from the x axis, and between the limits of 180 and +180 )? Correct answer: 26.1098 . Explanation:
sheriff (aas3563) HW1-a li (55140)
tan =
s2y s2x
2.
s2y = arctan s 2x
96.3841 cm = arctan 196.659 cm
= 26.1098 . 009 10.0 points ~ ~ ~ D, ~ and E ~ are shown in Vectors A, B, C, the figure. For convenience, the tails of each vector are arbitrarily located at (0,0). y A 5 4 C 3 2 1 x 0 1 E 2 3 D 4 5 5 3 B 1 0 1 2 3 4 5 Identify ~ = A ~ B ~ C ~ D ~ E ~, R 1. y 5 4 3 2 1 0 1 2 3 4 5 5
