What do you mean by resolution of a vector? What are the resolved parts called? Discuss a method to resolve a vector into two vectors in given direction?

Resolution of a vector is the process of splitting up the vector into two or more vectors so that combined effect of splitted vectors is same as that of given vector. The vectors into which a given vector is splitted is called component of vector. The components of a vector are also vectors.

Resolution of a vector is the process of splitting up the vector into

Resolution of a vector is the process of splitting up the vector into


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A motor boat is racing towards north at 25 km/hr and the water current in the river is 10 km/hr in the direction 60° east of south. Find the resultant velocity of the boat.

Let <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre> be the velocities of boat and river water respectively.

Let <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre> be the resultant velocity of boat.

Boat heads towards north and water current is in the direction <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre> east of south.

Therefore, the angle between <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre> is 120 degree.

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#6 {main}</pre> is ,

   open vertical bar straight v with rightwards arrow on top close vertical bar space equals space square root of straight v subscript straight b superscript 2 plus straight v subscript straight r superscript 2 plus 2 straight v subscript straight b straight v subscript straight r cosθ end root

       equals square root of left parenthesis 25 right parenthesis squared plus left parenthesis 10 right parenthesis squared plus 2 cross times 25 cross times 10 space cos space 120 end root

       equals square root of 625 plus 100 minus 250 end root space
equals space square root of 475 space
equals space 21.8 space km divided by hr 

Let the resultant velocity vector subtend an angle straight alpha with the north.

Then,

           tan space straight alpha space equals space fraction numerator straight v subscript straight r space sinθ over denominator straight v subscript straight b plus straight v subscript straight r cosθ end fraction

space space space space space space space space equals fraction numerator 10 space sin 120 degree over denominator 25 plus 10 space cos space 120 degree end fraction

                  equals fraction numerator 5 square root of 3 over denominator 20 end fraction

This implies,
                     straight alpha equals 23.4 degree

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Two forces have magnitude in the ratio 5:12. If the angle between them is 90°, they give the resultant of magnitude 26N. Find the magnitude of resultant force if the angle between them is changed to 120°.

Let magnitude of two forces be F1and F2.

Here,      straight F subscript 1 over straight F subscript 2 equals 5 over 12 

rightwards double arrow      space space space straight F subscript 2 equals 12 over 5 straight F subscript 1 equals 2.4 straight F subscript 1     ...(1)

The resultant of stack straight F subscript 1 with rightwards arrow on top space and space stack straight F subscript 2 with rightwards arrow on top is 26N when angle between them is .

Therefore,

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#6 {main}</pre>

                      equals straight F subscript 1 squared plus straight F subscript 2 squared              ...(2)

Substituting, straight F subscript 2 equals 2.4 space straight F subscript 1 in equation (2),

      left parenthesis 26 right parenthesis squared equals straight F subscript 1 squared plus straight F subscript 2 squared equals straight F subscript 1 squared plus left parenthesis 2.4 right parenthesis squared straight F subscript 1 squared

space space space space space space space space space equals 6.76 straight F subscript 1 squared
 
     left parenthesis 26 right parenthesis squared equals left parenthesis 2.6 right parenthesis squared straight F subscript 1 squared

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#6 {main}</pre>

∴  straight F subscript 2 equals 2.4 space space space space space straight F subscript 1 equals 24 straight N

Let R be the magnitude of the resultant when the angle between the two forces is 120 degree comma

∴                    straight R squared equals straight F subscript 1 squared plus straight F subscript 2 squared plus 2 straight F subscript 1 straight F subscript 2 space cos space 120 degree

                          equals straight F subscript 1 squared plus straight F subscript 2 squared minus straight F subscript 1 straight F subscript 2

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#6 {main}</pre>

i.e.,                   straight R equals 20.88, is the magnitude of the resultant between the vectors. 
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Two vectors acting in opposite direction have resultant 7 units and if they act at angle 90° to each other, the resultant is 13 units. Calculate the magnitude of the two vectors.

Let A and B be the magnitude of two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top and let A>B.

When straight A with rightwards arrow on top space and space straight B with rightwards arrow on top act in opposite direction, the magnitude of resultant is A - B

Now,

               A - B  = 7 [Given]              ...(1)

When straight A with rightwards arrow on top space and space straight B with rightwards arrow on top act perpendicular to each other, then their resultant is,

                         square root of straight A squared plus straight B squared end root space equals space 13

rightwards double arrow              straight A squared plus straight B squared equals 169                   ...(2)

Now,             left parenthesis straight A squared plus straight B squared right parenthesis minus left parenthesis straight A minus straight B right parenthesis squared equals 2 thin space straight A thin space straight B

∴                    2 AB space equals space 169 minus left parenthesis 7 right parenthesis squared equals 120

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#6 {main}</pre>

                             equals 169 plus 120 equals 289

or                 space space straight A plus straight B space equals space 17                    ...(3)

On solving equation (1) and (3), we get

                A = 12 and B = 5


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Show that there is one and only one set of resolved components in a particular direction.


Let a given vector straight R with rightwards arrow on top be resolved in 

Let a given vector  be resolved in Hence both the sets of componen
Hence both the sets of components are identical. There is one and only one set of resolved components in a particular direction.

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