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Determine if various combinations of vectors are defined or not.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Given the vectors $\\boldsymbol{v}$, $\\boldsymbol{w}$, $\\boldsymbol{u}$ in $\\mathbb{R}^3$, state whether the following quantities are scalars (real numbers), vectors (elements of $\\mathbb{R}^3$) or undefined.
\nIn this question, the symbol $\\cdot$ denotes the inner product and $\\times$ always denotes the cross product.
", "advice": "1. $\\boldsymbol{(v\\cdot w)\\cdot u}$ is undefined as $\\boldsymbol{v\\cdot w}$ is a scalar and we cannot take the inner product of a scalar with the vector $\\boldsymbol{u}$.
\n2. $\\boldsymbol{(v\\cdot w) u}$ is a vector and is a scalar multiple of $\\boldsymbol{u}$ as $\\boldsymbol{v \\cdot w}$ is a scalar.
\n3. $\\boldsymbol{(v \\cdot w)\\times u}$ is undefined as $\\boldsymbol{v\\cdot w}$ is a scalar and the cross product is only defined between vectors.
\n4. $\\boldsymbol{(v\\times w)\\times u}$ is a vector as $\\boldsymbol{v \\times w}$ and $\\boldsymbol{u}$ are vectors and the cross product between vectors produces a vector.
\n5. $\\boldsymbol{(v\\times w)\\cdot u}$ is a scalar as $\\boldsymbol{v \\times w}$ and $\\boldsymbol{u}$ are vectors and the inner or dot product is between vectors and produces a scalar.
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