Given Information:
Two vertices of an equilateral triangle are (0,0) and (3, √3).

Solution:

Step 1: Determine the distance between the two given vertices:
The distance between two points (x1, y1) and (x2, y2) is given by the distance formula:

d = √((x2-x1)^2 + (y2-y1)^2)

Using this formula, we can find the distance between (0,0) and (3, √3):

d = √((3-0)^2 + (√3-0)^2)
= √(9 + 3)
= √12
= 2√3

Step 2: Determine the midpoint of the line joining the given vertices:
The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula:

Midpoint = ((x1+x2)/2, (y1+y2)/2)

Using this formula, we can find the midpoint of the line joining (0,0) and (3, √3):

Midpoint = ((0+3)/2, (0+√3)/2)
= (3/2, √3/2)

Step 3: Determine the third vertex of the equilateral triangle:
To find the third vertex, we need to rotate the midpoint 60 degrees counterclockwise around one of the given vertices.

Step 3.1: Determine the angle of rotation:
Since the triangle is equilateral, all angles are 60 degrees. We need to rotate the midpoint 60 degrees counterclockwise.

Step 3.2: Apply the rotation formula:
To rotate a point (x, y) counterclockwise by an angle θ, the new coordinates (x', y') can be found using the following formulas:

x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)

In our case, the angle of rotation is 60 degrees and the midpoint is (3/2, √3/2). So, the third vertex coordinates will be:

x' = (3/2)*cos(60) - (√3/2)*sin(60)
= (3/2)*(1/2) - (√3/2)*(√3/2)
= 3/4 - 3/4
= 0

y' = (3/2)*sin(60) + (√3/2)*cos(60)
= (3/2)*(√3/2) + (√3/2)*(1/2)
= (3√3)/4 + (√3)/4
= (4√3)/4
= √3

Therefore, the coordinates of the third vertex are (0, √3).

Conclusion:
The coordinates of the third vertex of the equilateral triangle, given that two vertices are (0,0) and (3, √3), are (0, √3).