a*c=√3x-y

b*c=x+√3y

a*c=b*c，

√3x-y=x+√3y

(√3- 1)x=(√3+ 1)y

x=(√3+ 1)/(√3- 1)y

x=(2+√3)y

The modulus is the root number 2 √ (x 2+y 2) = √ 2.

x^2+y^2=2

(7+2√3)y^2+y^2=2

(4+√3)y^2= 1

y^2= 1/(4+√3)=(4-√3)/ 13

y = √( 4-√3)/ 13 =(√3- 1)/√ 13

x= (√3+ 1)√ 13

Coordinates of vector c ((√3+ 1)√ 13, (√3- 1)/√ 13) or (-(√3+ 1)√ 13.