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Find the angles that vector D = ( 2.0 i ^ 4.0 j ^ + k ^ ) m makes with the x -, y -, and z - axes.

θ i = 64.12 ° , θ j = 150.79 ° , θ k = 77.39 °

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Show that the force vector D = ( 2.0 i ^ 4.0 j ^ + k ^ ) N is orthogonal to the force vector G = ( 3.0 i ^ + 4.0 j ^ + 10.0 k ^ ) N .

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Assuming the + x -axis is horizontal to the right for the vectors in the previous figure, find the following vector products: (a) A × C , (b) A × F , (c) D × C , (d) A × ( F + 2 C ) , (e) i ^ × B , (f) j ^ × B , (g) ( 3 i ^ j ^ ) × B , and (h) B ^ × B .

a. −119.98 k ^ , b. −173.2 k ^ , c. + 93.69 k ^ , d. −413.2 k ^ , e. + 39.93 k ^ , f. −30.09 k ^ , g. + 149.9 k ^ , h. 0

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Find the cross product A × C for (a) A = 2.0 i ^ 4.0 j ^ + k ^ and C = 3.0 i ^ + 4.0 j ^ + 10.0 k ^ , (b) A = 3.0 i ^ + 4.0 j ^ + 10.0 k ^ and C = 2.0 i ^ 4.0 j ^ + k ^ , (c) A = −3.0 i ^ 4.0 j ^ and C = −3.0 i ^ + 4.0 j ^ , and (d) C = −2.0 i ^ + 3.0 j ^ + 2.0 k ^ and A = −9.0 j ^ .

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For the vectors in the earlier figure, find (a) ( A × F ) · D , (b) ( A × F ) · ( D × B ) , and (c) ( A · F ) ( D × B ) .

a. 0, b. 173,194, c. + 199,993 k ^

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(a) If A × F = B × F , can we conclude A = B ? (b) If A · F = B · F , can we conclude A = B ? (c) If F A = B F , can we conclude A = B ? Why or why not?

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Additional problems

You fly 32.0 km in a straight line in still air in the direction 35.0 ° south of west. (a) Find the distances you would have to fly due south and then due west to arrive at the same point. (b) Find the distances you would have to fly first in a direction 45.0 ° south of west and then in a direction 45.0 ° west of north. Note these are the components of the displacement along a different set of axes—namely, the one rotated by 45 ° with respect to the axes in (a).

a. 18.4 km and 26.2 km, b. 31.5 km and 5.56 km

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Rectangular coordinates of a point are given by (2, y ) and its polar coordinates are given by ( r , π / 6 ) . Find y and r .

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If the polar coordinates of a point are ( r , φ ) and its rectangular coordinates are ( x , y ) , determine the polar coordinates of the following points: (a) (− x , y ), (b) (−2 x , −2 y ), and (c) (3 x , −3 y ).

a. ( r , φ + π / 2 ) , b. ( 2 r , φ + 2 π ) , (c) ( 3 r , φ )

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Vectors A and B have identical magnitudes of 5.0 units. Find the angle between them if A + B = 5 2 j ^ .

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Starting at the island of Moi in an unknown archipelago, a fishing boat makes a round trip with two stops at the islands of Noi and Poi. It sails from Moi for 4.76 nautical miles (nmi) in a direction 37 ° north of east to Noi. From Noi, it sails 69 ° west of north to Poi. On its return leg from Poi, it sails 28 ° east of south. What distance does the boat sail between Noi and Poi? What distance does it sail between Moi and Poi? Express your answer both in nautical miles and in kilometers. Note: 1 nmi = 1852 m.

d PM = 33.12 nmi = 61.34 km , d NP = 35.47 nmi = 65.69 km

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An air traffic controller notices two signals from two planes on the radar monitor. One plane is at altitude 800 m and in a 19.2-km horizontal distance to the tower in a direction 25 ° south of west. The second plane is at altitude 1100 m and its horizontal distance is 17.6 km and 20 ° south of west. What is the distance between these planes?

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Show that when A + B = C , then C 2 = A 2 + B 2 + 2 A B cos φ , where φ is the angle between vectors A and B .

proof

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Four force vectors each have the same magnitude f . What is the largest magnitude the resultant force vector may have when these forces are added? What is the smallest magnitude of the resultant? Make a graph of both situations.

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Questions & Answers

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
who is Newton?
John Reply
scientist
Jeevan
a scientist
Peter
that discovered law of motion
Peter
ok
John
but who is Isaac newton?
John
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
elo
how?
Qhoshe
what is a scalar quantity
Peter Reply
scalar: are quantity have numerical value
muslim
is that a better way in defining scalar quantity
Peter
thanks
muslim
quantity that has magnitude but no direction
Peter
upward force and downward force lift
adegboye Reply
upward force and downward force on lift
adegboye
hi
Etini
hi
elo
hy
Xander
Hello
Jux_dob
hi
Peter
Helo
Tobi
what's the answer? I can't get it
Rachel Reply
what is the question again?
Sallieu
What's this conversation?
Zareen
what is catenation? and give examples
sununu
How many kilometres in 1 mile
Nessy
1.609km in 1mile
Faqir
what's the si unit of impulse
Iguh Reply
The Newton second (N•s)
Ethan
what is the s. I unit of current
Roland Reply
Amphere(A)
imam
thanks man
Roland
u r welcome
imam
the velocity of a boat related to water is 3i+4j and that of water related to earth is i-3j. what is the velocity of the boat relative to earth.If unit vector i and j represent 1km/hour east and north respectively
Pallavi Reply
what is head to tail rule?
kinza Reply
Explain Head to tail rule?
kinza
what is the guess theorem
Monu Reply
viva question and answer on practical youngs modulus by streching
Akash Reply
send me vvi que
rupesh
a car can cover a distance of 522km on 36 Liter's of petrol, how far can it travel on 14 liter of petrol.
Isaac
yoo the ans is 193
Joseph
whats a two dimensional force
Jimoh Reply
what are two dimensional force?
Ahmad
Where is Fourier Theorem?
Atul Reply
what is Boyle's law
Amoo Reply
Boyle's law state that the volume of a given mass of gas is inversely proportion to its pressure provided that temperature remains constant
Abe
how do I turn off push notifications on this crap app?
Huntergirl
what is the meaning of in.
CHUKWUMA Reply
In means natural logarithm
Elom
is dea graph for cancer caliper experiment using glass block?
Bako
why the ship created in search way?
Mershak
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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