U can find the answer on the TEM books, for example Prof. Graef's book.  The attached file is the corresponding part. Also, you can calculate the distance/spacing of both direct vector and reciprocal vector by using PTCLab. Download here: http://sourceforge.net/projects/tclab/

The vector S = ha* + kb* + lc* defines the hkl nodes of the reciprocal lattice of a crystal. Its module d*(hkl) represents the reciprocal of the interplanar spacing d(hkl) of the hkl family of lattice planes, i.e. d(hkl) =1/d*(hkl). However your question is not properly given because if the crystallographic reference system is not orthogonal (as in your hexagonal case) directions of the real space like yours [1 1 -2 0], [ 1 -1 0 0], [0 0 0 1] are, in general, not perpendicular to a hkl family of lattice planes; consequently those directions are not parallel to interplanar spacings. In your case only [0 0 0 1] is perpendicular to the (001) family of planes; in this case one has simply d(001) = c. With a hexagonal indexing (hkil) do calculations of d(hkl) the index i = -(h+k) must be omitted.

The easiest way is to use the metric tensor G, which is defined by the scalar products of the basis vectors, cf. the already by Gu Xinfu mentioned Marc DeGraef book .

                              h

1/d^2= (hkl) G^-1   ( k )

                              l

whereas the square of the length of lattice vecor R=[uvw] is given by

                        u

|R|^2=[uvw] G [ v ]

                        w

A critical problem G. Ferraris already mentioned: You should transfer the lattice and reciprocal lattice vectors from Miller-Bravais and Weber indexing into Miller indexing- For planes the description already G. Ferraris displayed. For R=[uvtw] it is a bit more difficult. Simply use the Weber matrix M:

                 U

[uvtw]=M [ V ]

                W

with M=

 2   -1   0

-1    2   0

-1   -1   0

  0    0   3

To derive [UVW] - the Miller indexing - from [uvtw] you need to inverse M

                        u

[UVW] = M^-1 [ v ]

                         t

                        w

M^-1=

1   0   -1   0

0   1   -1   0

0   0    0   1

The big advantage: you can transfer or calculate all by the application of G (or G^-1).

These equations are independent on the crystal system.

Narendra Bandaru; i reply here also to your private message. It is my feeling that you (as many others posing basic questions of crystallography through ResearchGate) had very little or even no courses of basic crystallography. In fact the calculation of interplanar spacings, d(hkl), is a very basic exercise. In similar sistuations it becomes very difficult, not to say impossible, to help you. Anyway, to obtain the data you are looking for, you need the cell parameters of the two compounds forming the interface. Then using the formula relating d(hkl) to (a, b, c, alpha, beta, gamma) and (hkl)  (BASIC CRYSTALLOGRAPHY!) you can calculate d(hkl). Since you need values only for few simple directions, the calculations can even be done manually in a short time.  Finally, this is for tutors and teachers: students/researchers cannot deal with crystalline materials without a minimum of general preparation on crystallography. 

This is obviously a general problem in ResearchGate. It seems that there is a strong believe it is easier to ask a question than to go to a library (and read a book) or even use Wikipedia (which is also not a real compensation of a fundamental course in crystallography or any other science).

http://calistry.org/calculate/spacingTetragonalLattice

Hello,

Each direction represents the normal vector of a plane. So, we can find the miller indices of the planes (hkil). Then, there is a formula which states the relation between d spacing and h,k,i,l for the hexagonal unit cell.  You can find the equation for d spacing here:

http://geoweb3.princeton.edu/research/MineralPhy/xtalgeometry.pdf

Hope it helps!

Narendra Bandaru

In this video tutorial, I have explained in detail how and how to calculate lattice constants for cubic and orthorhombic structures from the XRD data using OriginLab software. In the case you want to further ask about it, please do comment on the specific video, I'll respond to it shortly. I have provided the practice as well as calculations files here. Thanks

https://youtu.be/Hr3S_f4lktw