If coordinates of points A, B, C and D are respectively (3, 5, -3), (2, 3, -1), (1, 2, 3) and (3, 5, 7), then :

Question

Accepted Answer

To determine the relationship between the line segments $\overline{AB}$ and $\overline{CD}$, we need to analyze their direction vectors and the angle between them.

1. **Direction Vectors:**
   - The direction vector of $\overline{AB}$ is obtained by subtracting the coordinates of point A from point B:
     $$
     \overrightarrow{AB} = (2 - 3, 3 - 5, -1 + 3) = (-1, -2, 2)
     $$
   - The direction vector of $\overline{CD}$ is obtained by subtracting the coordinates of point C from point D:
     $$
     \overrightarrow{CD} = (3 - 1, 5 - 2, 7 - 3) = (2, 3, 4)
     $$

2. **Dot Product and Perpendicularity:**
   - Two vectors are perpendicular if their dot product is zero. Let's calculate the dot product of $\overrightarrow{AB}$ and $\overrightarrow{CD}$:
     $$
     \overrightarrow{AB} \cdot \overrightarrow{CD} = (-1)(2) + (-2)(3) + (2)(4) = -2 - 6 + 8 = 0
     $$
   - Since the dot product is zero, $\overline{AB}$ is perpendicular to $\overline{CD}$.

3. **Explanation of Incorrect Options:**
   - **Option 2: $\overline{AB}||\overline{CD}$:** For two vectors to be parallel, one must be a scalar multiple of the other. The vectors $(-1, -2, 2)$ and $(2, 3, 4)$ are not scalar multiples of each other, so they are not parallel.
   - **Option 3: Angle between $\overline{AB}$ and $\overline{CD}$ is $\frac{\pi}{3}$:** The angle between two vectors is given by the formula:
     $$
     \cos \theta = \frac{\overrightarrow{AB} \cdot \overrightarrow{CD}}{\|\overrightarrow{AB}\| \|\overrightarrow{CD}\|}
     $$
     Since the dot product is zero, $\cos \theta = 0$, which corresponds to an angle of $\frac{\pi}{2}$, not $\frac{\pi}{3}$.
   - **Option 4: Angle between $\overline{AB}$ and $\overline{CD}$ is $\frac{\pi}{6}$:** Similarly, as explained above, the angle is $\frac{\pi}{2}$, not $\frac{\pi}{6}$.

In conclusion, the correct answer is Option 1: $\overline{AB}\perp\overline{CD}$, as the dot product of their direction vectors is zero, confirming that they are perpendicular.