Cross product of 3 dimensional vectors
$a=<a_1,a_2,a_3>, b=<b_1,b_2,b_3>$ can be defined as
$c=<a_2 b_3 - a_3 b_2, a_3 b_1 - a_1 b_3, a_1 b_2 - a_2 b_1>$
Length of the vector $c$ can be defined as:
$|c|=\sqrt{\sum_{i=1}^{3} c_{i}^{2}}$
or equivalently:
$|c|=|a| |b| sin (\phi)$, where $\phi$ is the angle between $a$ and $b$.
Now when I compute cross product in vector form: $a\times b = a \times (b1+b2) = a \times b1 + a \times b2$ holds ($b1, b2 \in \mathbb{R}^{3}$, $b=b1+b2$), but when doing this with formula for scalar: $|c|=|a| |b| sin (\phi)$, it linearity does not hold.
Example:
$a=<1, 2, 3>$
$b=<-3, 3, 1>$
$b1=<-2, 1, 0>$
$b2=<-1, 2, 1>$
I calculate value of sinus based on dot product: $sin \phi = \sqrt{(1-cos^{2}\phi)} = \sqrt{(1 - \frac {<a, b>^{2}}{|a|^{2}|b|^{2}})}$
$|a \times (b1 + b2)| = |a \times b| = |a| |b| sin(a, b) = 3.74 * 4.36 * 0.93 = 15.2$
$|a \times b1 + a \times b2)| = ~ 15.3$
What have I misunderstood? What is the problem? What can cause it?
Somewhere in your calculations you are making a formula error. I don't know where, since you haven't shown your calculations. But the formulas you've shown here are correct. However, the values are not. Both values are $\sqrt{230}\approx 15.16575$.
To see this, note that $$a \times b\phantom{2} = (-7, -10, \phantom{-}9)\\a \times b1 = (-3, \phantom{1}-6, \phantom{-}5)\\ a\times b2 = (-4, \phantom{1}-4, \phantom{-}4)$$ So obviously $a \times b = (a\times b1) + (a\times b2)$. Now $$|a \times b| = \sqrt{(-7)^2 + (-10)^2 + 9^2}= \sqrt{230}$$ This alone tells where the problem is. You have mis-calculated either $a\times b1, a\times b2$, their sum, or $|a\times b|$, since $\sqrt{230} \approx 15.16575 < 15.3$.
But for completeness, note that $$|a|^2 = 14, |b|^2 = 19, \langle a, b\rangle^2 = 6^2 = 36$$ So $$\cos^2\phi = \frac{36}{14\cdot 19}\\\sin^2\phi = 1 - \frac{36}{14\cdot 19} = \frac{230}{14\cdot 19}$$ Therefore $$|a\times b|^2 = |a|^2|b|^2\sin^2\phi = 14\cdot 19\cdot\frac{230}{14\cdot 19} = 230$$ giving $|a\times b| = \sqrt{230}$, just as before.