Geometry

3D Coordinate Geometry

3D Coordinate Geometry - Skew Lines

         

Calculate the distance between the following two lines: r=x−13=y−4−1=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} .

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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 cos⁡θ?\cos \theta?

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

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If the following two lines are not skew, what is the value of the non-zero variable u:u: {l1:x2=y3=z−22l2:x2=y4=z−uu?\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}

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Find the distance between the following two lines:

{l1:x=−y=8−zl2: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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