Geometry

3D Coordinate Geometry

3D Coordinate Geometry: Level 2 Challenges

         

Point PP is some point on the surface of the sphere (x1)2+(y+2)2+(z3)2=1. {(x-1)}^{2}+{(y+2)}^{2}+{(z-3)}^{2}=1 . What is the shortest possible distance between PP and O=(0,0,0)?O=(0, 0, 0)?

If the two lines x1k=y+12=z,x+23=1y=z+2k\begin{aligned} \frac {x-1 }{k } &=\frac { y+1 }{2 } =z,\\ \\ \\ \frac {x+2 }{-3 } & =1-y =\frac{z+2}{k} \end{aligned} are perpendicular to each other, then what is the value of k?k?

The points AA, BB, PP, and QQ all lie on one line, with A=(3,5,8).A=(-3, 5, 8).

Point PP lies in the xyxy-plane between AA and BB such that the distance from AA to PP is twice the distance from PP to BB. Furthermore, QQ sits on the zz-axis such that the distance from AA to QQ is twice the distance from QQ to BB.

If B=(x,y,z),B=(x,y,z), what is the value of x+y+z?x+y+z?

An infinite column is centered along the zz-axis. It has a square cross-section of side length 10. It is cut by the plane 4x7y+4z=25.4x - 7y + 4z = 25.

What is the area of the surface cut?

If the point Q=(a,b,c)Q=(a, b, c) is the reflection of the point P=(6,2,3)P=(-6, 2, 3) about the plane 3x4y+5z9=0,3x-4y+5z-9=0, determine the value of a+b+c.a+b+c.

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