Prove that `|axxb|^2 =a^2b^2 - (a.b)^2`

Prove that |axxb^2 =a^2b^2 - (a.b)^2

If vec a and vec b are two vectors,then prove that (vec a xxvec b)^(2)=a^(2)b^(2)-(a.b)^(2)

Prove that |[(b+c)^2, a^2, bc],[(c+a)^2, b^2, ca],[(a+b)^2, c^2, ab]|=(a-b)(b-c)(c-a)(a+b+c)(a^2+b^2+c^2)

Prove that: (a-b)^(2)=a^(2)-2ab+b^(2)

Prove that matrix [((b^(2)-a^(2))/(a^(2)+b^(2)),(-2ab)/(a^(2)+b^(2))),((-2ab)/(a^(2)+b^(2)),(a^(2)-b^(2))/(a^(2)+b^(2)))] is orthogonal.

If (a^(2)-b^(2))sin theta+2ab cos theta=a^(2)+b^(2) then prove that tan theta=(a^(2)-b^(2))/(2ab)

Using properties of determinants, prove that |[a^2, bc, ac+c^2] , [a^2+ab, b^2, ac] , [ab, b^2+bc, c^2]| = 4a^2b^2c^2

Prove that |[b^2+c^2,ab,ac],[ab,c^2+a^2,bc],[ca,cb,a^2+b^2]| is always positive for real a, b and c.

Prove that det[[a^(2),2ab,b^(2)b^(2),a^(2),2ab2ab,b^(2),a^(2)]]=(a^(3)+b^(3))^(2)