Question

True/False: If $a$ and $b$ are scalars such that $a \mathbf{u}+b \mathbf{v}=\mathbf{0}$, then $\mathbf{u}$ and $\mathbf{v}$ are parallel vectors.

Answer 1

False. If $a\mathbf{u} + b\mathbf{v} = \mathbf{0}$, it means that the vectors $\mathbf{u}$ and $\mathbf{v}$ are linearly dependent. However, this does not necessarily mean that the vectors are parallel.

For example, consider the vectors $\mathbf{u} = \begin{pmatrix} 1 \ 0 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 0 \ 1 \end{pmatrix}$. These vectors are linearly dependent, since we can express one of them as a multiple of the other:

$$\mathbf{v} = 2 \mathbf{u}$$

However, these vectors are not parallel, since they do not point in the same direction.

In general, two vectors $\mathbf{u}$ and $\mathbf{v}$ are parallel if and only if there exists a scalar $k$ such that $\mathbf{v} = k\mathbf{u}$. If this condition is not met, the vectors are not parallel.