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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

a length of copper wire was measured to be 50m with an uncertainty of 1cm, the thickness of the wire was measured to be 1mm with an uncertainty of 0.01mm, using a micrometer screw gauge, calculate the of copper wire used
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What is the answer please
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how many cm makes 1 inches
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2.5
omwoyo
2.54cm=1inche
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how do we convert from m/s to km/hr
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Explain why polarization does not occur in sound
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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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