Two vectors acting in opposite directions have a resultant of 10 units. If they act at right angles to each other, the resultant is 50 units. Calculate the magnitudes of the two vectors
Answers
Answer:
Hence, magnitude of two vectors are 40 and 30 .
Explanation:
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 .
Explanation:
kindly refer to attachment
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