ok heres how you do the question...
you know the cube roots of unity for 1+w+w^2=0
Now this also forms the vertices of an equilateral triange.
so you can see now that alpha, beta and phi are also these vertices if you let them be free vectors.
since 1, w and w^2 are the roots or vertices, we can equate alpha = 1, beta = w and phi = w^2.
the problem is alpha^2+beta^2+phi^2 = Alpha beta + beta phi + alpha phi
therefore : 1^2 + w^2 + (w^2)^2 = 1w + w(w^2) + 1w^2 (1)
note that in cube roots of unity w^3 = 1, w^4 = w etc.
therefore continuing with equation 1;
LHS: 1+w^2+w^4
= 1 + w + w^2
= 0
RHS = w + w^3 + w^2
= 1 + w + w^2
= 0
= LHS
Of course there might be other ways to hit this question but you can use this.
Anyways im up for a challenge of hard 3u maths and 4u maths stuff so please send me an email of your question if you got a hard one and i shall send you the document of it hand written scanned on comp, so its easier to read. email is
silveroak555@hotmail.com hope we'll get this together
PS: dont send me anything more than complex and graphing till the next term cause i havent done them haha. oh and w = omega in the question. and no im not some 4eye asian nerd who looks like hes gonna break but i am asian so yeah.