The correct answer is: B. orthogonal
Two vectors are orthogonal if their dot product is zero. The dot product of two vectors $\mathbf{u}$ and $\mathbf{v}$ is defined as follows:
$$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3$$
In this case, the two vectors are given by:
$$\mathbf{u} = (1, 1, 1)$$
and
$$\mathbf{v} = (1, a, a^2)$$
Therefore, the dot product of these two vectors is:
$$\mathbf{u} \cdot \mathbf{v} = 1 \cdot 1 + 1 \cdot a + 1 \cdot a^2 = a^2 + 2a + 1$$
Since this is not equal to zero, the two vectors are not orthogonal.
To determine if the two vectors are parallel, we can check if they have the same direction. The direction of a vector is given by its unit vector, which is found by dividing the vector by its magnitude. The magnitude of a vector is found by taking the square root of the sum of the squares of its components.
In this case, the unit vectors of the two vectors are given by:
$$\hat{\mathbf{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|} = \frac{(1, 1, 1)}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{1}{\sqrt{3}} (1, 1, 1)$$
and
$$\hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{(1, a, a^2)}{\sqrt{1^2 + a^2 + a^4}} = \frac{1}{\sqrt{1 + a^2 + a^4}} (1, a, a^2)$$
Since the unit vectors of the two vectors are not equal, the two vectors are not parallel.
Therefore, the only option that remains is B. orthogonal.