Question

two vectors acting in opposite direction have a resultant of 10 unit. if they act at right angles to each other the resultant is 50 unit calculate magnitude of 2 vectors

Answer 1

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$\textbf { your answer is :- }$

Let, two vector be A and B.

since, Two vectors acting in opposite direction have a resultant of 10 unit
it means angle between these two vectors is 180° .

so, √A^2+B^2+2ABcos180° = 10

=> A^2+B^2+2ABcos180° = 100

=> A^2+B^2+2AB×-1 = 100 [ cos180°= -1]

=> A^2+B^2-2AB = 100

=> (A-B)^2 = 100

=> A-B = √100

=> A-B = 10 ......(1)


also, since if they act at right angles to each other the resultant is 50 unit

so, √A^2+B^2+2ABcos90° = 50

=> A^2+B^2+2ABcos90° = 2500

=> A^2+B^2 = 2500 [cos90° = 0]

=> (A-B)^2+2AB = 2500

=> 10^2 + 2AB = 2500 ( from 1 )

=> 2AB = 2500 - 100

=> 2AB = 2400

=> AB = 2400/2

=> AB = 1200

=> B = 1200/A

put this value in equation (1) , we get

A-B = 10

=> A - 1200/A= 10

=> A^2 -10A -1200 = 0

=> A^2 - 40A+30A -1200 = 0

=> A(A-40) + 30(A-40) = 0

=> (A-40) (A+30) = 0

=> A = 40 or -30

=> A = 40 [ since magnitude never negative ]

Now, put the value of A in equation (1) , we get

A-B = 10

=> 40-B = 10

=> B = 40-10

=> B = 10


Hence, magnitude of two vectors are 40 and 30 .


$\textbf { HOPE IT HELP YOU }$

Answer 2

Answer:

Well, the two vectors are A and B.

as two alternate vectors have a unit effect of 10 units

means that the angle between these two vectors is 180 °.

thus, √A ^ 2 + B ^ 2 + 2ABcos180 ° = 10

=> A ^ 2 + B ^ 2 + 2ABcos180 ° = 100

=> A ^ 2 + B ^ 2 + 2AB × -1 = 100 [cos180 ° = -1]

=> A ^ 2 + B ^ 2-2AB = 100

=> (A-B) ^ 2 = 100

=> A-B = √100

=> A-B = 10 ...... (1)

and, as they do at right angles each result is 50 units

thus, √A ^ 2 + B ^ 2 + 2ABcos90 ° = 50

=> A ^ 2 + B ^ 2 + 2ABcos90 ° = 2500

=> A ^ 2 + B ^ 2 = 2500 [cos90 ° = 0]

=> (A-B) ^ 2 + 2AB = 2500

=> 10 ^ 2 + 2AB = 2500 (from 1)

=> 2AB = 2500 - 100

=> 2AB = 2400

=> AB = 2400/2

=> AB = 1200

=> B = 1200 / A

put this number in equation (1), we find

A-B = 10

=> A - 1200 / A = 10

=> A ^ 2 -10A -1200 = 0

=> A ^ 2 - 40A + 30A -1200 = 0

=> A (A-40) + 30 (A-40) = 0

=> (A-40) (A + 30) = 0

=> A = 40 or -30

=> A = 40 [as size has never been negative]

Now, enter the value of A in equation (1), we get it

A-B = 10

=> 40-B = 10

=> B = 40-10

=> B = 10

Therefore, the magnitude of the two vectors is 40 and 30.

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