Answer
The correct answer is
(b) $\left( {{\bf{u}} + {\bf{w}}} \right)\cdot\left( {{\bf{v}} \times {\bf{w}}} \right)$.
Work Step by Step
Consider (a) ${\bf{v}} \times \left( {{\bf{u}} + {\bf{w}}} \right)$.
By properties of the cross product we have
${\bf{v}} \times \left( {{\bf{u}} + {\bf{w}}} \right) = {\bf{v}} \times {\bf{u}} + {\bf{v}} \times {\bf{w}}$
Since the cross products of two vectors is a vector, and the sum of two vectors is a vector, we conclude that ${\bf{v}} \times \left( {{\bf{u}} + {\bf{w}}} \right)$ is a vector.
Consider (b) $\left( {{\bf{u}} + {\bf{w}}} \right)\cdot\left( {{\bf{v}} \times {\bf{w}}} \right)$.
Since the sum of two vectors is a vector, so $\left( {{\bf{u}} + {\bf{w}}} \right)$ is a vector. Since the cross products of two vectors is a vector, so $\left( {{\bf{v}} \times {\bf{w}}} \right)$ is a vector. Since the dot product of two vectors is a scalar, we conclude that $\left( {{\bf{u}} + {\bf{w}}} \right)\cdot\left( {{\bf{v}} \times {\bf{w}}} \right)$ is a scalar.
Consider (c) $\left( {{\bf{u}} \times {\bf{w}}} \right) + \left( {{\bf{w}} - {\bf{v}}} \right)$.
Since $\left( {{\bf{u}} \times {\bf{w}}} \right)$ is a vector and $\left( {{\bf{w}} - {\bf{v}}} \right)$ is a vector, the sum $\left( {{\bf{u}} \times {\bf{w}}} \right) + \left( {{\bf{w}} - {\bf{v}}} \right)$ is a vector.
Hence, the correct answer is (b) $\left( {{\bf{u}} + {\bf{w}}} \right)\cdot\left( {{\bf{v}} \times {\bf{w}}} \right)$.
