What about these

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 $\hat { j }$

Easy Math Editor

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Comments

$cosx + cosy + cosz = a$

$\cos^2x + \cos^2y + \cos^2z + 2\cos x\cos y + 2\cos y\cos z + 2 \cos x\cos y = a^2$

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, $\cos^2x + \cos^2y + \cos^2z + 2\cos x\cos y + 2\cos y\cos z + 2 \cos x\cos y +$

$\sin^2x + \sin^2y + \sin^2z + 2\sin x\sin y + 2\sin y\sin z + 2 \sin x\sin y = a^2 + b^2$

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

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

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

$0 = a^2 + b^2 \implies a = 0\quad \& \quad b = 0$

Siddhartha Srivastava - 6 years, 1 month ago

Thanks for this man.

Raghav Vaidyanathan - 6 years, 1 month ago

@Raghav Vaidyanathan @Ronak Agarwal @Mvs Saketh

Tanishq Varshney - 6 years, 1 month ago

The answer to q 1 is very easy: $a=b=0$ .

Raghav Vaidyanathan - 6 years, 1 month ago

Ans 2 q2 seems to be zero too.

Raghav Vaidyanathan - 6 years, 1 month ago

can u explain in brief the first part

Tanishq Varshney - 6 years, 1 month ago

@Tanishq Varshney – for first one The simplest way is to assume all the three angle differences equal, which ofcourse then must be 120 degree, so just consider 1 , w ,w^2 , add em up

Mvs Saketh - 6 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$