Past exam of the mathematical methods I course of the University of Oxford / 2021 resit / 9 / i

Use $\mathbf{w}=-\mathbf{u}-\mathbf{v}$ and bilinearity and antisymmetry of the cross product:Consequently,If $\mathbf{u},\mathbf{v}$ are linearly independent, this linear span is the plane through the origin with normal vector $\mathbf{u}\times \mathbf{v}$. If they are dependent, all three nonzero vectors lie on one line through the origin, and their linear span has dimension one. For example, $\mathbf{u}=\mathbf{v}=(1,0,0)$ and $\mathbf{w}=(-2,0,0)$ obey the hypothesis but span only a line. Nonzero vectors alone do not force a two-dimensional linear span.