Consider $\mathbf{u}, \mathbf{v}, \mathbf{w}$ vectors.
If $\mathbf{u}$ and $\mathbf{v}$ are orthogonal, that is $\mathbf{u} \cdot \mathbf{v} = 0$, and the vectors $\mathbf{v}$ and $\mathbf{w}$ are parallel, that is $\mathbf{v} = c\mathbf{w}$ for some $c$, then by transitivity we can conclude that
$\mathbf{u} \cdot c\mathbf{w} = 0$
EDIT: Can we conclude this through some principle? Is it the transitive property?
We can certainly conclude it, but not by the transitive property.
The transitive property simply states that if $a = b$ and $b=c$, then $a=c$.
What we are using instead is the substitution property of equality: if $a=b$ and we have an expression involving $a$, we can replace $a$ with $b$ without affecting the expression. This substitution property is much stronger than transitivity.
EDIT: As an example, consider the relation $u \sim v$ when $\|u\| = \|v\|$. It's easy to see that $\sim$ is an equivelance relation, and in particular, satisfies the transitive property. But for arbitrary vectors $u,v,w$ with $\|v\|=\|w\|$, you can't go around replacing $v$ with $w$ in formulas like $u\cdot v$ and expect to get the same answer, namely, $u \cdot v = u \cdot w$.