Question

Two vectors acting in the opposite directions have a resultant of 10 units. If they act at right angles to each other, then the resultant is 50 units. Calculate the magnitude of two vectors.

Answer 1

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 .

I HOPE IT HELPS U A LOT....

TQ FOR ASKING THIS QUESTION....

#BE BRAINLY