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A skater glides along a circular path of radius 5.00 m in clockwise direction. When he coasts around one-half of the circle, starting from the west point, find (a) the magnitude of his displacement vector and (b) how far he actually skated. (c) What is the magnitude of his displacement vector when he skates all the way around the circle and comes back to the west point?

a. 10.00 m, b. 5 π m , c. 0

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A stubborn dog is being walked on a leash by its owner. At one point, the dog encounters an interesting scent at some spot on the ground and wants to explore it in detail, but the owner gets impatient and pulls on the leash with force F = ( 98.0 i ^ + 132.0 j ^ + 32.0 k ^ ) N along the leash. (a) What is the magnitude of the pulling force? (b) What angle does the leash make with the vertical?

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If the velocity vector of a polar bear is u = ( −18.0 i ^ 13.0 j ^ ) km / h , how fast and in what geographic direction is it heading? Here, i ^ and j ^ are directions to geographic east and north, respectively.

22.2 km/h, 35.8 ° south of west

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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.

Vector G has magnitude 10.0. Its projection in the x y plane is between the positive x and positive y directions, at an angle of 45 degrees from the positive x direction. The angle between vector G and the positive z direction is 60 degrees. Vector H has magnitude 15.0. Its projection in the x y plane is between the negative x and positive y directions, at an angle of 30 degrees from the positive y direction. The angle between vector H and the positive z direction is 450 degrees.
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A diver explores a shallow reef off the coast of Belize. She initially swims 90.0 m north, makes a turn to the east and continues for 200.0 m, then follows a big grouper for 80.0 m in the direction 30 ° north of east. In the meantime, a local current displaces her by 150.0 m south. Assuming the current is no longer present, in what direction and how far should she now swim to come back to the point where she started?

240.2 m, 2.2 ° south of west

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A force vector A has x - and y -components, respectively, of −8.80 units of force and 15.00 units of force. The x - and y -components of force vector B are, respectively, 13.20 units of force and −6.60 units of force. Find the components of force vector C that satisfies the vector equation A B + 3 C = 0 .

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Vectors A and B are two orthogonal vectors in the xy -plane and they have identical magnitudes. If A = 3.0 i ^ + 4.0 j ^ , find B .

B = −4.0 i ^ + 3.0 j ^ or B = 4.0 i ^ 3.0 j ^

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For the three-dimensional vectors in the following figure, find (a) G × H , (b) | G × H | , and (c) G · H .

Vector G has magnitude 10.0. Its projection in the x y plane is between the positive x and positive y directions, at an angle of 45 degrees from the positive x direction. The angle between vector G and the positive z direction is 60 degrees. Vector H has magnitude 15.0. Its projection in the x y plane is between the negative x and positive y directions, at an angle of 30 degrees from the positive y direction. The angle between vector H and the positive z direction is 450 degrees.
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Show that ( B × C ) · A is the volume of the parallelepiped, with edges formed by the three vectors in the following figure.

Vector G has magnitude 10.0. Its projection in the x y plane is between the positive x and positive y directions, at an angle of 45 degrees from the positive x direction. The angle between vector G and the positive z direction is 60 degrees. Vector H has magnitude 15.0. Its projection in the x y plane is between the negative x and positive y directions, at an angle of 30 degrees from the positive y direction. The angle between vector H and the positive z direction is 450 degrees.

proof

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Challenge problems

Vector B is 5.0 cm long and vector A is 4.0 cm long. Find the angle between these two vectors when | A + B | = 3.0 cm and | A B | = 3.0 cm .

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What is the component of the force vector G = ( 3.0 i ^ + 4.0 j ^ + 10.0 k ^ ) N along the force vector H = ( 1.0 i ^ + 4.0 j ^ ) N ?

G = 2375 17 9792

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The following figure shows a triangle formed by the three vectors A , B , and C . If vector C is drawn between the midpoints of vectors A and B , show that C = C / 2 .

Vectors A, B and C form a triangle. Vector A points up and right, vector B starts at the head of A and points down and right, and vector C starts at the head of B, ends at the tail of A and points to the left. Vector C prime is parallel to vector C and connects the midpoints of vectors A and B.
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Distances between points in a plane do not change when a coordinate system is rotated. In other words, the magnitude of a vector is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle φ to become a new coordinate system S , as shown in the following figure. A point in a plane has coordinates ( x , y ) in S and coordinates ( x , y ) in S .

(a) Show that, during the transformation of rotation, the coordinates in S are expressed in terms of the coordinates in S by the following relations:

{ x = x cos φ + y sin φ y = x sin φ + y cos φ .

(b) Show that the distance of point P to the origin is invariant under rotations of the coordinate system. Here, you have to show that

x 2 + y 2 = x 2 + y 2 .

(c) Show that the distance between points P and Q is invariant under rotations of the coordinate system. Here, you have to show that

( x P x Q ) 2 + ( y P y Q ) 2 = ( x P x Q ) 2 + ( y P y Q ) 2 .

Two coordinate systems are shown. The x y coordinate system S, in red, has positive x to to the right and positive y up. The x prime y prime coordinate system S prime, in blue, shares the same origin as S but is rotated relative to S counterclockwise an angle phi. Two points, P and Q are shown. Point P’s x coordinate in frame S is shown as a dashed line from P to the x axis, drawn parallel to the y axis. Point P’s y coordinate in frame S is shown as a dashed line from P to the y axis, drawn parallel to the x axis. Point P’s x prime coordinate in frame S prime is shown as a dashed line from P to the x prime axis, drawn parallel to the y prime axis. Point P’s y prime coordinate in frame S prime is shown as a dashed line from P to the y prime axis, drawn parallel to the x prime axis.

proof

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Questions & Answers

what is the guess theorem
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viva question and answer on practical youngs modulus by streching
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a car can cover a distance of 522km on 36 Liter's of petrol, how far can it travel on 14 liter of petrol.
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what are two dimensional force?
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what is Boyle's law
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Boyle's law state that the volume of a given mass of gas is inversely proportion to its pressure provided that temperature remains constant
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what is the meaning of in.
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In means natural logarithm
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is dea graph for cancer caliper experiment using glass block?
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identity of vectors?
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what is defined as triple temperature
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Triple temperature is the temperature at which melting ice and boiling water are at equilibrium
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a tire 0.5m in radius rotate at constant rate 200rev/min. find speed and acceleration of small lodged in tread of tire.
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100
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yes
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_2/3 ÷34
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what larminar flow
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smooth or regular flow
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Hii
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scalar field define with example
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what is displacement
Isaac Reply
the change in the position of an object in a particular direction is called displacement
Noor
The physical quantity which have both magnitude and direction are known as vector.
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good
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Describe vector integral?
Malik
define line integral
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Examples on how to solve terminal velocity
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Practice Key Terms 6

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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