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

Explanation

Let, the unit vectors along `vecA, vecB and vecR " be"" " hata , hatb and hatc` respectively. Then
`vecA=Ahata,vecB=Bhatb,vecR=Rhatc and Ahata+Bhatb=Rhatc`
` or, A(vec(OP))/(OP)+B(vec(OQ))/(OS)=R(vec(OS))/(OS)` [Fig .2.62].

If `hatn` is the unit vector along the line CD,
`vec(OP)xxhatn=OP sin alphahatm=ONhatm,`
`vec(OQ)xxhatn=ONhatm, vec(OS)xxhatn=ONhatm`
where `hatm` is a unit vector perpendicular to the plane of the vectors.
Then , `A/(OP)vec(OP)xxhatn+B/(OQ)vec(OQ)xxhatn=R/(OS)vec(OS)xxhatn`
`or, A/(OP) ONhatm+B/(OQ)hatm=R/(OS)ONhatm`
`or, A/(OP)+B/(OQ)=R/(OS)`.