# 2.4 Products of vectors  (Page 8/16)

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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. $5\pi \phantom{\rule{0.2em}{0ex}}\text{m}$ , c. 0

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 $\stackrel{\to }{F}=\left(98.0\stackrel{^}{i}+132.0\stackrel{^}{j}+32.0\stackrel{^}{k}\right)\text{N}$ along the leash. (a) What is the magnitude of the pulling force? (b) What angle does the leash make with the vertical?

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

22.2 km/h, $35.8\text{°}$ south of west

Find the scalar components of three-dimensional vectors $\stackrel{\to }{G}$ and $\stackrel{\to }{H}$ in the following figure and write the vectors in vector component form in terms of the unit vectors of the axes.

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 $30\text{°}$ 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, $2.2\text{°}$ south of west

A force vector $\stackrel{\to }{A}$ 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 $\stackrel{\to }{B}$ are, respectively, 13.20 units of force and −6.60 units of force. Find the components of force vector $\stackrel{\to }{C}$ that satisfies the vector equation $\stackrel{\to }{A}-\stackrel{\to }{B}+3\stackrel{\to }{C}=0$ .

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

$\stackrel{\to }{B}=-4.0\stackrel{^}{i}+3.0\stackrel{^}{j}$ or $\stackrel{\to }{B}=4.0\stackrel{^}{i}-3.0\stackrel{^}{j}$

For the three-dimensional vectors in the following figure, find (a) $\stackrel{\to }{G}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{H}$ , (b) $|\stackrel{\to }{G}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{H}|$ , and (c) $\stackrel{\to }{G}·\stackrel{\to }{H}$ .

Show that $\left(\stackrel{\to }{B}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{C}\right)·\stackrel{\to }{A}$ is the volume of the parallelepiped, with edges formed by the three vectors in the following figure.

proof

## Challenge problems

Vector $\stackrel{\to }{B}$ is 5.0 cm long and vector $\stackrel{\to }{A}$ is 4.0 cm long. Find the angle between these two vectors when $|\stackrel{\to }{A}+\stackrel{\to }{B}|=\phantom{\rule{0.2em}{0ex}}3.0\phantom{\rule{0.2em}{0ex}}\text{cm}$ and $|\stackrel{\to }{A}-\stackrel{\to }{B}|=\phantom{\rule{0.2em}{0ex}}3.0\phantom{\rule{0.2em}{0ex}}\text{cm}$ .

What is the component of the force vector $\stackrel{\to }{G}=\left(3.0\stackrel{^}{i}+4.0\stackrel{^}{j}+10.0\stackrel{^}{k}\right)\text{N}$ along the force vector $\stackrel{\to }{H}=\left(1.0\stackrel{^}{i}+4.0\stackrel{^}{j}\right)\text{N}$ ?

${G}_{\perp }=2375\sqrt{17}\approx 9792$

The following figure shows a triangle formed by the three vectors $\stackrel{\to }{A}$ , $\stackrel{\to }{B}$ , and $\stackrel{\to }{C}$ . If vector ${\stackrel{\to }{C}}^{\prime }$ is drawn between the midpoints of vectors $\stackrel{\to }{A}$ and $\stackrel{\to }{B}$ , show that ${\stackrel{\to }{C}}^{\prime }=\stackrel{\to }{C}\text{/}2$ .

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 $\phi$ to become a new coordinate system ${\text{S}}^{\prime }$ , as shown in the following figure. A point in a plane has coordinates ( x , y ) in S and coordinates $\left({x}^{\prime },{y}^{\prime }\right)$ in ${\text{S}}^{\prime }$ .

(a) Show that, during the transformation of rotation, the coordinates in ${\text{S}}^{\prime }$ are expressed in terms of the coordinates in S by the following relations:

$\left\{\begin{array}{c}{x}^{\prime }=x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\phi +y\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\phi \\ {y}^{\prime }=\text{−}x\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\phi +y\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\phi \end{array}.$

(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

$\sqrt{{x}^{2}+{y}^{2}}=\sqrt{{{x}^{\prime }}^{2}+{{y}^{\prime }}^{2}}.$

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

$\sqrt{{\left({x}_{P}-{x}_{Q}\right)}^{2}+{\left({y}_{P}-{y}_{Q}\right)}^{2}}=\sqrt{{\left({{x}^{\prime }}_{P}-{{x}^{\prime }}_{Q}\right)}^{2}+{\left({{y}^{\prime }}_{P}-{{y}^{\prime }}_{Q}\right)}^{2}}.$

proof

Is there any calculation for line integral in scalar feild?
what is thrust
when an object is immersed in liquid, it experiences an upward force which is called as upthrust.
Phanindra
@Phanindra Thapa No, that is buoyancy that you're talking about...
Shii
thrust is simply a push
Shii
it is a force that is exerted by liquid.
Phanindra
what is the difference between upthrust and buoyancy?
misbah
The force exerted by a liquid is called buoyancy. not thrust. there are many different types of thrust and I think you should Google it instead of asking here.
Sharath
hey Kumar, don't discourage somebody like that. I think this conversation is all about discussion...remember that the more we discuss the more we know...
festus
thrust is an upward force acting on an object immersed in a liquid.
festus
uptrust and buoyancy are the same
akanbi
Shii
a Thrust is simply a push
Shii
how did astromers neasure the mass of earth and sun
wats the simplest and shortest formula to calc. for order of magnitude
papillas
Distinguish between steamline and turbulent flow with at least one example of each
what is newtons first law
It state that an object in rest will continue to remain in rest or an object in motion will continue to remain in motion except resultant(unbalanced force) force act on it
Gerald
Thanks Gerald Fokumla
Theodore
Gerald
it states that a body remains in its state of rest or uniform motion unless acted upon by resultant external force.
festus
it that a body continues to be in a state of rest or in straight line in a motion unless there is an external force acting on it
Usman
derive the relation above
formula for find angular velocity
w=v^2/r
Eric
Why satellites don't fall on earth? Reason?
because space doesn't have gravity
Evelyn
satellites technically fall to earth but they travel parallel to earth so fast that they orbit instead if falling(plus the gravity is also weaker in the orbit). its a circular motion where the centripetal force is the weight due to gravity
Kameyama
Exactly everyone what is gravity?
the force that attrats a body towards the center of earth,or towards any other physical body having mass
hina
That force which attracts or pulls two objects to each other. A body having mass has gravitational pull. If the object is bigger in mass then it's gravitational pull would be stronger.For Example earth have gravitational pull on other objects that is why we are pulled by earth.
Abdur
Gravity is the force that act on a on body to the center of the earth.
Aguenim
what are the application of 2nd law
It's applicable when determining the amount of force needed to make a body to move or to stop a moving body
festus
coplanar force system
how did you get 7.50times
6
Mharsheeraz
what is a frame of reference
0.88
Mharsheeraz
0.88
Iize
The system of geometric axis in relation to which measurement of Size, Position, or , Motion can be made. It has two types; 1) Inertial Reference Frame 2) Non Inertial Reference Frame
Abdur
what is science
What is Matter?
Lloyd
mu
matter is anything having some mass and occupies some volume
Debi
Evans
is anything that occuple space
Diamond
I'm new here...I wana askng u how can I prepare any typ of test of atomic energy
gull
What is inertia of bank curve
Sunny
I wanna ask the different between coloumbs law and gravitational law of force
Femi
how do you got 27.8 m/s? please explain