1. **Problem Statement:** Given points A(3,6) and B(9,4) on the line of action of force $\vec{F}$, with position vectors $\vec{r}$ for A and $\vec{r_1}$ for B, determine which statement about the moment $\vec{M}_O$ about origin O is incorrect. 2. **Recall the moment formula:** The moment of a force about a point O is given by $$\vec{M}_O = \vec{r} \times \vec{F}$$ where $\vec{r}$ is the position vector from O to the point of application of the force. 3. **Important rule:** The moment is independent of the point chosen on the line of action of the force. This means the moment about O can be calculated using any point on the line of action. 4. **Vectors involved:** - $\vec{r} = (3,6)$ (position vector of A) - $\vec{r_1} = (9,4)$ (position vector of B) - Vector from B to A: $\overrightarrow{BA} = \vec{r} - \vec{r_1} = (3-9,6-4) = (-6,2)$ - Vector from A to B: $\overrightarrow{AB} = \vec{r_1} - \vec{r} = (9-3,4-6) = (6,-2)$ 5. **Check each statement:** - a) $\vec{M}_O = (3,6) \times \vec{F} = \vec{r} \times \vec{F}$ — Correct by definition. - b) $\vec{M}_O = [\vec{r} + \overrightarrow{BA}] \times \vec{F} = (3,6) + (-6,2) = (-3,8) \times \vec{F}$ Since $\vec{r} + \overrightarrow{BA} = \vec{r_1}$, this equals $\vec{r_1} \times \vec{F}$, which is valid. - c) $\vec{M}_O = (9,4) \times \vec{F} = \vec{r_1} \times \vec{F}$ — Correct by the same reasoning. - d) $\vec{M}_O = [\vec{r} + \overrightarrow{AB}] \times \vec{F} = (3,6) + (6,-2) = (9,4) \times \vec{F}$ This equals $\vec{r_1} \times \vec{F}$, which is correct. 6. **Identify the incorrect statement:** Statement b) uses $\vec{r} + \overrightarrow{BA} = \vec{r_1}$, which is correct, so b) is correct. All statements a), b), c), and d) are correct except statement a) if the force $\vec{F}$ is not applied at point A but along the line AB. However, since the problem asks for the incorrect statement and all given statements are correct by vector addition and moment properties, the only exception is statement a) if $\vec{F}$ is not applied at A. But since the problem states $\vec{F}$ acts along line AB, the moment about O can be calculated using either point A or B or any point on AB. Therefore, the only incorrect statement is **a)** if $\vec{F}$ is not applied exactly at A but along the line. **Final answer:** The incorrect statement is **a) $\vec{M}_O = (3,6) \times \vec{F}$**.