The three vectors $vec{a}, vec{b} and vec{c}$ are mutually orthogonal vectors in $R^{3}$ such that the triad $(\vec{a}, \vec{b}, \vec{c})$ is a right–handed triad and $|\vec{a}| = 1, |\vec{b}| = 2$ and $|\vec{c}| = 3$.
The vectors $\vec{u}, \vec{v}$ and $\vec{w}$ are given in terms of the basis vectors $\vec{a}, \vec{b}$ and $\vec{c}$ as follows:
$\vec{u} = 3\vec{a} + 3\vec{b} − \vec{c}$
$\vec{v} = \vec{a} + 2\vec{b} + 3\vec{c}$
$\vec{w} = \vec{a} + \vec{b} + \vec{c}.$
Determine $(\vec{u} ∧ \vec{v}) · \vec{w}$.
How can I find the cross product of two vectors which do not make use of unit vectors?
No, you don't compute the cross product and dot product separately, but make use of the scalar triple product immediately.
So \begin{align*} (u\times v)\cdot w&=[u,v,w]\\ &=[3a + 3b − c,a + 2b + 3c, a + b + c]\\ &=\det\begin{pmatrix}3&3&-1\\1&2&3\\1&1&1\end{pmatrix}[a,b,c]\\ &=4[a,b,c]\\ &=4\lvert a\rvert\cdot\lvert b\rvert\cdot\lvert c\rvert\quad\text{since }(a,b,c)\text{ right-handed triad}\\ &=\dots \end{align*}