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The resultant of two vectors vec A and v...

The resultant of two vectors `vec A and vec B` is perpendicular to the vector `vec A` and its magnitude is equal to half the magnitude of vector `vec B`. What is the angle between `vec A and vec B` ?

Text Solution

Verified by Experts

Let angle between `vec A and vec B` be `theta`
Let `vec C=vecA +vecB`
Let `alpha` be the angle between `vecC and vecA`, then we can write the following :
`tan alpha =(B sin theta)/(A+B cos theta)`
Here `alpha=90^(@)`
`rArr tan 90^(@) =(B sin theta)/(A+B cos theta)`
`A+B cos theta=0`
Magnitude of vector `vec C ` can be written as follows:
`rArr C=sqrt(A^(2)+B^(2)+2AB cos theta)`
Here `C=B//2`
`rArr (B^(2))/(4)=A^(2)+B^(2)+2AB cos theta `
We can substitute from equation (i) : `B cos theta =-A`
`rArr (B^(2))/(4)=A^(2)+B^(2)-2A^(2)`
`rArr A^(2)=(3)/(4) B^(2) rArr (A)/(B) =(sqrt3)/(2)`
Substituting the above result in equation (i) again , we get the following :
`cos theta =-(sqrt3)/(2) rArr theta =150^(@)`
Hence the angle between the two given vectors is `150^(@)`
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