Problem 35

To analyze each option step-by-step:

Option A: If $\overrightarrow{a} + \overrightarrow{b} = \overrightarrow{0}$, then we can rearrange this equation to $\overrightarrow{a} = -\overrightarrow{b}$. This implies that vector $\overrightarrow{a}$ is a scalar multiple of vector $\overrightarrow{b}$, which means $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$. Therefore, statement A is correct.

Option B: Given vector $\overrightarrow{AB} \neq \overrightarrow{AC}$, it implies that the vectors from point $A$ to points $B$ and $C$ are different. Since vectors are defined by both magnitude and direction, a difference in either would mean that points $B$ and $C$ are distinct, i.e., they do not coincide. Thus, statement B is correct.

Option C: A vector pointing east-south $70^{\circ}$ and a vector pointing north-west $20^{\circ}$ are described. The direction of the first vector is $70^{\circ}$ away from the east towards the south, and the second vector is $20^{\circ}$ away from the north towards the west. These directions are opposite to each other, indicating that the vectors are collinear but in opposite directions. Hence, statement C is correct.

Option D: If $\overrightarrow{a}$ and $\overrightarrow{b}$ are parallel vectors, it means that $\overrightarrow{a}$ can be expressed as a scalar multiple of $\overrightarrow{b}$, i.e., $\overrightarrow{a} = \lambda\overrightarrow{b}$. This relationship only ensures directionality and does not impose any condition on the magnitudes of $\overrightarrow{a}$ and $\overrightarrow{b}$ being equal. Therefore, the magnitudes $|\overrightarrow{a}|$ and $|\overrightarrow{b}|$ can be different, making statement D incorrect.

Given the analysis above, the incorrect conclusion among the options is:

$\boxed{\text{D}}$.