Question

Determine which of the following are defined for nonzero vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\). Explain your reasoning. (a) \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})\) (b) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\) (c) \(\mathbf{u} \cdot \mathbf{v}+\mathbf{w}\) (d) \(\|\mathbf{u}\| \cdot(\mathbf{v}+\mathbf{w})\)

Short answer

Option (a) and option (d) are defined for non-zero vectors \mathbf{u}, \mathbf{v}, and \mathbf{w} while option (b) and option (c) are not.

Step-by-step solution

Analyse option (a)

For the expression \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})\), it's the dot product between vector \mathbf{u} and the result of addition of vectors \mathbf{v} and \mathbf{w}. Since both the dot product and vector addition are defined for non-zero vectors, this expression is valid.

Analyse option (b)

The expression \((\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\) is not defined because it implies multiplying a scalar, which is the result of the dot product, by a vector, with standard vector operations, this is undefined.

Analyse option (c)

Looking at \(\mathbf{u} \cdot \mathbf{v}+\mathbf{w}\), it implies adding a scalar, the result of the dot product of \mathbf{u} and \mathbf{v}, to a vector \mathbf{w}. This is undefined with standard vector operations.

Analyse option (d)

In \(\|\mathbf{u}\| \cdot(\mathbf{v}+\mathbf{w})\), the magnitude of a vector \mathbf{u} is being multiplied by the vector result of the addition of vectors \mathbf{v} and \mathbf{w}. Multiplying a scalar (in this case the magnitude) by a vector is defined, and so is adding two vectors. Therefore this expression is valid.