The answer in my book starts with :
Let r = xa + yb + z(a x b). [x,y and z are variables whereas a and b are the vectors which are mutually perpendicular/]
And ultimately, they find out x, y and z by doing dot product of r and a,b and (a x b). (Which I understand)
What I fail to understand is : why do they take r to be xa + yb + z(a x b) ? What do each of "xa" , "xb" and "y(a x b)" signify?
On
The three-dimensional vectors $a$ and $b$ are orthonormal vectors. Thus their cross product $$a×b=|a||b|\sin\phi,$$ with $\phi$ being the angle between them, is orthogonal to the two of them. Also, since $\phi=π/2,$ we have that $|a×b|=1,$ so that the triad of vectors also forms an orthonormal basis, which will then span the whole of $\mathrm R^3$ -- i.e., every vector in this space may be written as a linear combination of $a,b,a×b.$ This is what they've done, taking $x,y,z$ to be variable scalars.
It seems that $\mathbf{r}$ is a 3D vector. In order to represent the 3D vector in the $(x,y,z)$ system all we need are 3 mutually orthogonal vectors. The simplest of these are pointing along the axes, and these are commonly known as the $(i,j,k)$ unit vectors. BUT any three orthogonal vectors will do.
So we have the information $a$ and $b$ are already orthgonal. Now we need one more vector for this 3D system to be well represented. We know the cross product $a\times b$ by definition generates a new vector perp. to original two.
Therefore we have our three orthogonal components in this 3D system, represented by vectors: $(a,b,a\times b)$. Using this we can find our variables $x,y,z$ (which are not the same concept by the way as we used to describe our system i.e. the $(x,y,z)$ co-ordinate system).