Geometry

3D Coordinate Geometry

3D Coordinate Geometry - Skew Lines

         

Calculate the distance between the following two lines: r=x13=y41=z+14 r = \frac{x-1}{3}=\frac{y-4}{-1}=\frac{z+1}{4} and {x=4+λy=1z=8+2λ. \begin{cases} x = 4 +\lambda \\ y = 1 \\ z= 8 + 2\lambda \end{cases} .

In the above regular octahedron ABCDEF,ABCDEF, MM and NN are the midpoints of BC\overline{BC} and CD,\overline{CD}, respectively. If θ\theta is the angle between NF and MA,\overrightarrow{NF} \text{ and } \overrightarrow{MA}, what is the value of cosθ?\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:u: {l1:x2=y3=z22l2:x2=y4=zuu?\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:

{l1:x=y=8zl2:x=y and z=10.\begin{cases} l_1: x=-y=8-z \\\\ l_2: x=y \text{ and } z=-10. \end{cases}

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