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3D Coordinate Geometry

Append a z-axis to the 2-dimensional plane and conquer the realm of 3-dimensional space.

Skew Lines

Calculate the distance between the following two lines: \[ r = \frac{x-1}{3}=\frac{y-4}{-1}=\frac{z+1}{4} \] and \[ \begin{cases} x = 4 +\lambda \\ y = 1 \\ z= 8 + 2\lambda \end{cases} .\]

In the above regular octahedron \(ABCDEF,\) \(M\) and \(N\) are the midpoints of \(\overline{BC}\) and \(\overline{CD},\) respectively. If \(\theta\) is the angle between \(\overrightarrow{NF} \text{ and } \overrightarrow{MA}, \) what is the value of \(\cos \theta?\)

Count the number of pairs of edges in the above cuboid that are skew.

If the following two lines are not skew, what is the value of the non-zero variable \(u:\) \[\begin{cases} l_1: \frac{x}{2}=\frac{y}{3}=\frac{z-2}{2} \\\\ l_2 :\frac{x}{2}=\frac{y}{4}=\frac{z-u}{u}? \end{cases}\]

Find the distance between the following two lines:

\[\begin{cases} l_1: x=-y=8-z \\\\ l_2: x=y \text{ and } z=-10. \end{cases}\]

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