unit vector in 2D space = draw circle with center as (0,0) with radius 1 unit
all points lying on the circle are unit vectors.
unit vector in 3D space = draw hollow sphere with center as (0,0,0) with radius 1 unit
all points lying on the surface area of sphere are unit vectors.
so in any case distance from (0,0) to that vector is 1.
now we can define anyyy vector which is arbitrary , as an unit vector.
say, (7,4) in 2D space,
( 7/sqrt[7^2+4^2] , 4/sqrt[7^2+4^2] ) is the unit vector corresponding to (7,4)
for recheck find distance from (0,0) ; it will be 1
now given A a matrix, let (p,q) is its E. vector
now the corresponding unit E. vector is (r,s) column vect. say.
r=p/sqrt(p^2+q^2)
s=q/sqrt(p^2+q^2)
so, xT.A.x= xT.L.x (as Ax=Lx) =L.xT.x =L.(r,s).TR(r,s) =L.(r^2+s^2) =L.1= L
so maximum e.val. is the answer.