Q1) If \(cos(x-y)+cos(y-z)+cos(z-x)=\frac{-3}{2}\) and \(cosx+cosy+cosz=a\) and \(sinx+siny+sinz=b\). Find \(a\) and \(b\)

Q2) Two non-conducting infinite wires have shapes as shown above. The linear charge densities of wire (1) and wire (2) are

\(x\quad if\quad x\leq -a~ and~ x\geq a\\ \\ 0\quad if\quad -a<x<a)\)

and

\(y\quad if\quad y\leq -a~ and~ y\geq a\\ \\ 0\quad if\quad -a<y<a)\)

respectively where \((x,y)\) are coordinates of points on the wire. An electron is released from origin, the unit vector along the direction of velocity of the electron just after its release will be in terms of \(\hat { i } \) and \(\hat { j } \)

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TopNewest\( cosx + cosy + cosz = a \)

\( \cos^2x + \cos^2y + \cos^2z + 2\cos x\cos y + 2\cos y\cos z + 2 \cos x\cos y = a^2\)

Likewise, \( \sin^2x + \sin^2y + \sin^2z + 2\sin x\sin y + 2\sin y\sin z + 2 \sin x\sin y = b^2\)

Adding, \( \cos^2x + \cos^2y + \cos^2z + 2\cos x\cos y + 2\cos y\cos z + 2 \cos x\cos y + \)

\( \sin^2x + \sin^2y + \sin^2z + 2\sin x\sin y + 2\sin y\sin z + 2 \sin x\sin y = a^2 + b^2 \)

\( \implies 1 + 1 + 1 + 2(\cos x\cos y + \cos y\cos z + \cos x\cos y + \sin x\sin y + \sin y\sin z + \sin x\sin y) = a^2 + b^2 \)

\( \implies 3 + 2(\cos(x-y) + \cos(y-z) + \cos(z-x)) = a^2 + b^2 \)

\( \implies 3 + 2* \frac{-3}{2} = a^2 + b^2 \)

\( 0 = a^2 + b^2 \implies a = 0\quad \& \quad b = 0 \)

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Thanks for this man.

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The answer to q 1 is very easy: \(a=b=0\).

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Ans 2 q2 seems to be zero too.

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can u explain in brief the first part

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@Raghav Vaidyanathan @Ronak Agarwal @Mvs Saketh

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