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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

a particle projected from origin moving on x-y plane passes through P & Q having consituents (9,7) , (18,4) respectively.find eq. of trajectry.
rahul Reply
definition of inertia
philip Reply
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
charles
An inherent property by virtue of which the body remains in its pure state or initial state
Kushal
why current is not a vector quantity , whereas it have magnitude as well as direction.
Aniket Reply
why
daniel
the flow of current is not current
fitzgerald
bcoz it doesn't satisfy the algabric laws of vectors
Shiekh
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
Kushal
what is binomial theorem
Tollum Reply
hello are you ready to ask aquestion?
Saadaq Reply
what is binary operations
Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
Martan Reply
position, velocity and acceleration of vector
Manuel Reply
hi
peter
hi
daniel
hi
Vedisha
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
imam
hello
Lydia
hello Lydia.
Sackson
What is momentum
isijola
hello
Saadaq
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
I need the solving for this question
philip
is the eye the same like the camera
EDWIN Reply
I can't understand
Suraia
same here please
Josh
I think the question is that ,,, the working principal of eye and camera same or not?
Sardar
yes i think is same as the camera
muhammad
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
MUNGWA Reply
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
Victor
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
Goodness Reply
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
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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