Are reciprocal lattice basis vectors parallel to their corresponding real lattice basis vectors? So I had a question asking to find the angle between b* and (113) plane normal in a tetragonal unit cell. I know that the angle between two plane normals are given by:

My thinking was that if I know a direction that is parallel to b*. I could put that direction to the formula and calculate the angle.
I know that b*.b = 1
So my questions is if this means b* and b are parallel to each other and if I use [010] direction, will I make a mistake?
I also read from De Graeg, Structure of Materials that:

So this had me confused, are the reciprocal lattice basis vectors parallel to their corresponding real lattice basis vectors, perpendicular or neither of the two?
 A: 
are the reciprocal lattice basis vectors parallel to their corresponding
real lattice basis vectors, perpendicular or neither of the two?

No, in general they are neither.
Let $\mathbf{b}_1$, $\mathbf{b}_2$, $\mathbf{b}_3$ be the basis vectors of the real lattice,
and $\mathbf{b}^*_1$, $\mathbf{b}^*_2$, $\mathbf{b}^*_3$ the basis vectors of the reciprocal lattice.
According to Reciprocal lattice - Three dimension they are related by:
$$\begin{align}
\mathbf{b}^*_1&=\frac{2\pi}{V}\mathbf{b}_2\times\mathbf{b}_3 \\
\mathbf{b}^*_2&=\frac{2\pi}{V}\mathbf{b}_3\times\mathbf{b}_1 \\
\mathbf{b}^*_3&=\frac{2\pi}{V}\mathbf{b}_1\times\mathbf{b}_2
\end{align}$$
From the properties of the $\times$ product, it is clear that

*

*$\mathbf{b}^*_1$ is perpendicular to $\mathbf{b}_2$ and $\mathbf{b}_3$,

*$\mathbf{b}^*_2$ is perpendicular to $\mathbf{b}_3$ and $\mathbf{b}_1$,

*$\mathbf{b}^*_3$ is perpendicular to $\mathbf{b}_1$ and $\mathbf{b}_2$.

However, this usually does not mean that $\mathbf{b}^*_i$ is parallel to $\mathbf{b}_i$.
And it certainly doesn't mean $\mathbf{b}^*_i$ is perpendicular to $\mathbf{b}_i$.
This can already be seen in two dimensions.

It is also not hard to convince yourself about the same fact in three dimensions.
(But it is harder to draw, and therefore I didn't bother with that.)
A: 
The reciprocal lattice vector g with components (h,k,l) is
perpendicular to the plane with Miller index (hkl).

Yes.
Suppose $3$ non orthogonal vectors a, b and c, like $3$ lines from one of the vertex of a tetrahedron. Each $2$ of them determine a plane. In a bravais lattice, this is a cristallographic plane, with a long range symmetry. The cross product of these vectors generates a new vector perpendicular to this plane. And perpendicular to each of those vectors. If we define for example:$$\mathbf G_1 = \frac{\mathbf a \times \mathbf b}{(\mathbf a \times \mathbf b) \mathbf {.c}} \implies \mathbf G_1 \mathbf{.c} = 1\; \mathbf G_1 \mathbf{.b} = \mathbf G_1 \mathbf{.a} = 0$$
