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Prove that if two vectors have the same magnitude $v$ and make an angle $\theta$ , their sum has a magnitude $S=2v\cos\frac{1}{2}\theta$ and their difference is $D=2v\sin\frac{1}{2}\theta$ .

#Mechanics #Sam

Easy Math Editor

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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}`	$\boxed{123}$

Comments

If you have 2 vectors of magnitudes $a$ and $b$ with an angle $\theta$ between them, then their resultant has a magnitude of $\sqrt{a^2+b^2+2ab\times cos\theta}$ . (Try to prove this using the triangle law of vector addition).

Thus in your 1st case, the resultant is $\sqrt{v^2+v^2 + 2v^2 cos\theta} = \sqrt{2v^2(1+cos\theta)} = v \sqrt{2+2cos \theta} = v\sqrt{4 cos^2\frac{\theta}{2}} = 2v\times cos\frac{\theta}{2}$ .....

we used the result " $cos\bigl( \frac{\theta}{2}\bigr) = \sqrt{\frac{1+cos\theta}{2}}$ ", which is easy to prove using $cos\theta = \cos(\frac{\theta}{2}+\frac{\theta}{2})= cos^2 (\frac{\theta}{2}) -sin^2(\frac{\theta}{2})=2cos^2(\frac{\theta}{2})-1$

In the seconds case, only difference occurring is $\theta$ becomes $180^\circ -\theta$ hence in the half angle thing, it will become $90^\circ - cos(\frac{\theta}{2})$ and that will become $sin(\frac{\theta}{2})$ , giving the final result as
$2 u \times sin(\frac{\theta}{2})$

Aditya Raut - 6 years, 11 months ago

oh thanks for your contribution aditya!

samuel ayinde - 6 years, 11 months ago

go for vector addition

S=\sqrt { v^{ 2 }+v^{ 2 }+2v.v.\cos \theta } =\sqrt { 2v^{ 2 }+2v^{ 2 }\cos \theta } =\sqrt { 2v^{ 2 }(1+\cos \theta ) } =\sqrt { 2v^{ 2 }(2\cos ^{ 2 } \theta /2) } ............................changing1+\cos \theta to half angle form =\sqrt { 4v^{ 2 }\cos ^{ 2 } \theta /2 } =2v\cos \theta /2

change this to text. And do same for difference.

Rajeev Sharma - 6 years, 11 months ago

ok i'll try it. thanks @Rajeev sharma

samuel ayinde - 6 years, 11 months 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}`	$\boxed{123}$