Physics, asked by boago572, 1 year ago

The resultant of two vectors u and v is perpendicular to u. If |v|=2^1\2 |u| , show that the resultant of 2u and v is perpendicular to v.

Answers

Answered by sonuvuce
0

Answer:

If the angle between vectors \vec u and  \vec u is \theta then the angle made by the resultant with \vec u is given by

\boxed{\phi=\tan^{-1}\frac{v\sin\theta}{u+v\cos\theta}}

Since the angle \phi=90^\circ

Therefore,

u+v\cos\theta=0

\implies \cos\theta=-\frac{u}{v}

But given that

v=\sqrt{2}u

Thus

\cos\theta=-\frac{1}{\sqrt{2}}

\implies \cos\theta=\cos135^\circ

\implies \theta=135^\circ

Now if the magnitude of the vector \vec u is doubled then if the angle between \vec v and the resultant is \phi'

Then

\phi'=\tan^{-1}\frac{2u\sin135^\circ}{v+2u\cos135^\circ}

\implies \tan\phi'=\frac{2u\sin135^\circ}{v+2u\cos135^\circ}

Now the value of

v+2u\cos135^\circ

=v+2u\times (-\frac{1}{\sqrt{2}})

=v-\sqrt{2}u

=0          (As given in the question)

\implies \tan\phi'=\infty}

or, \phi'=90^\circ

Therefore, the resultant of the vector 2\vec u and \vec v is perpendicular to \vec v

Hope this helps.

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