Question

part d
compare the resulting vectors for u - v and u + (-1v). what can you conclude based on the comparison?

Explanation:

To solve this, we recall the definition of vector subtraction. By the definition of vector operations, subtracting a vector \(\mathbf{v}\) from \(\mathbf{u}\) (\(\mathbf{u}-\mathbf{v}\)) is equivalent to adding the negative of \(\mathbf{v}\) (which is \(- 1\times\mathbf{v}\) or \(-\mathbf{v}\)) to \(\mathbf{u}\) (\(\mathbf{u}+(- 1\mathbf{v})\)). So when we compare the resulting vectors of \(\mathbf{u}-\mathbf{v}\) and \(\mathbf{u}+(-1\mathbf{v})\), we will find that they are equal. This is because of the algebraic property of vector operations where \(\mathbf{u}-\mathbf{v}=\mathbf{u}+(-\mathbf{v})\) (and \(-\mathbf{v}\) is the same as \((- 1)\mathbf{v}\) in terms of scalar multiplication of vectors).

Answer:

The resulting vectors for \(\mathbf{u}-\mathbf{v}\) and \(\mathbf{u}+(-1\mathbf{v})\) are equal. We can conclude that subtracting a vector \(\mathbf{v}\) from a vector \(\mathbf{u}\) is equivalent to adding the scalar -multiplied vector \((- 1)\mathbf{v}\) to \(\mathbf{u}\), i.e., \(\mathbf{u}-\mathbf{v}=\mathbf{u}+(-1\mathbf{v})\).