In Biot-Savart law: what is magnitude and direction of $\mathbf{B}_C$? In Biot-Savart law we have :
$$\mathbf{B}_C=\int_C \Big(\dfrac{\mu_0}{4\pi} \Big)\Big(\dfrac{I\,d\vec{l} \times \mathbf{r}}{\|l\mathbf{r}\|l^3}\Big)
$$
Now what is magnitude and direction of $\mathbf{B}_C$ ?
Can I say that :
$$\|\mathbf{B}_C\|l=\int_C \Big(\dfrac{\mu_0}{4\pi} \Big)\Big(\dfrac{I\|d\vec{l}\|\,l \, \|\mathbf{r}\| \sin\alpha}{\|\mathbf{r}\|^3}\Big)$$
And also :
direction of magnetic field is direction of $(d\vec{l} \times \mathbf{r})$ ? How is it proved? 
 A: Neither of your conclusions is correct. Neither the magnitude nor the direction of the field can be determined a priori just from the integral expression without actually evaluating the integral (except in cases of exceptional symmetry). Most importantly, the direction of $d\vec{l} \times \mathbf{r}$ is ambiguous outside of the integral, since the directions of both $d\vec{l}$ and $\mathbf{r}$ vary as one integrates over the current path, so it makes no sense to claim that expression as the direction of $\mathbf{B}_C$. The only completely general way to determine the direction of $\mathbf{B}_C$ is to evaluate the Biot-Savart law integral component-wise to find the components of $\mathbf{B}_C$, their relative sizes telling you the direction. Sometimes symmetry of the current path tells you that components in a particular (set of) direction(s) must cancel, leaving only the orthogonal components, but this is by no means general. 
Because the direction of $d\vec{l} \times \mathbf{r}$ varies, the magnitudes of this vector at each point do not simply add, and so your proposed expression for the magnitude of $\mathbf{B}_C$ is also not correct. Instead, your expression is an upper bound on the magnitude of $\mathbf{B}_C$, since the combined magnitudes cannot be greater than the sum of the magnitudes.
