Sort each expression by what kind of object it produces ($u, v$ are vectors).

Vector: $u + v$, $3u$, $u \times v$ (cross product, in 3D)

Scalar: $u \cdot v$ (dot product), $\|u\|$ (norm)

Type-checking expressions, vector or scalar?, catches most linear-algebra errors before any arithmetic. Adding a scalar to a vector, for example, is as meaningless as adding seconds to metres.