Is the cross product or the scalar product implied if neither are explicitly used in an expression?

Question

For example, does the expression $ab$ imply $a \cdot b$ or $a \times b$ ,what about the expression $\vec a \vec b$ , does this imply $\vec a \cdot \vec b$ or $\vec a \times \vec b$ ? What if brackets are included: $\vec a (\vec b)$?

Answer 1

It could mean just a dyadic product, if $\vec{a} = a_x \hat{x} + a_y \hat{y} + a_z \hat{z}$ and $\vec{b} = b_x \hat{x} + b_y \hat{y} + b_z \hat{z}$, then

\begin{eqnarray} \vec{a}\vec{b} &=& (a_x \hat{x} + a_y \hat{y} + a_z \hat{z})(b_x \hat{x} + b_y \hat{y} + b_z \hat{z}) \\ &=& a_xb_x \hat{x}\hat{x} + a_xb_y\hat{x}\hat{y} + a_xb_z\hat{x}\hat{z} \\ && + a_yb_x \hat{y}\hat{x} + a_yb_y\hat{y}\hat{y} + a_yb_z\hat{y}\hat{z} \\ && +a_zb_x \hat{z}\hat{x} + a_zb_y\hat{z}\hat{y} + a_zb_z\hat{z}\hat{z} \\ \end{eqnarray}

This is just tensor product