# Vector Related problem If vec(a),vec(b) &vec(c0 are unit vector such that |vec(a)−vec(b)|^2+|vec(b)-vec(c)|^2+|vec(c)−vec(a)∣^2=9. How do I proceed

If $\stackrel{\to }{a},\stackrel{\to }{b}\mathrm{&}\stackrel{\to }{c}$ are unit vector such that ${|\stackrel{\to }{a}-\stackrel{\to }{b}|}^{2}+{|\stackrel{\to }{b}-\stackrel{\to }{c}|}^{2}+{|\stackrel{\to }{c}-\stackrel{\to }{a}|}^{2}=9$
Then the value $|2\stackrel{\to }{a}+5\stackrel{\to }{b}+5\stackrel{\to }{c}|=\mathrm{_}\mathrm{_}\mathrm{_}\mathrm{_}\mathrm{_}\mathrm{_}$
My approach is as follow
${|\stackrel{\to }{a}-\stackrel{\to }{b}|}^{2}+{|\stackrel{\to }{b}-\stackrel{\to }{c}|}^{2}+{|\stackrel{\to }{c}-\stackrel{\to }{a}|}^{2}={|\stackrel{\to }{a}|}^{2}+{|\stackrel{\to }{b}|}^{2}-2\stackrel{\to }{a}.\stackrel{\to }{b}+{|\stackrel{\to }{b}|}^{2}+{|\stackrel{\to }{c}|}^{2}-2\stackrel{\to }{b}.\stackrel{\to }{c}+{|\stackrel{\to }{c}|}^{2}+{|\stackrel{\to }{a}|}^{2}-2\stackrel{\to }{c}.\stackrel{\to }{a}=9$
$\left(\stackrel{\to }{a}.\stackrel{\to }{b}+\stackrel{\to }{b}.\stackrel{\to }{c}+\stackrel{\to }{c}.\stackrel{\to }{a}\right)=-\frac{3}{2}$
How do I proceed from here
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Kristin Myers
For convenience we shall take $a=\stackrel{\to }{a},b=\stackrel{\to }{b},c=\stackrel{\to }{c}$
$a\cdot b+b\cdot c+c\cdot a+\frac{3}{2}=0$
$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}|a+b+c{|}^{2}=0$
$|a+b+c|=0$
$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}a+b+c=0$
$|2a+5b+5c|=|2a-5a|=3$