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A rectangular solid consisting of 18 smaller cubes that are identical is positioned in the standard (x, y, z) coordinate system. Vertex M has coordinates of (-1, 3, 0) and point O on the y axis has points of (0, 3, 0). What are the coordinates of Vertex N?enter image description here

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    $\begingroup$ I would suggest adding a description of what you have tried/where you are stuck exactly. People will be more sympathetic to helping you then. Also, for clarity, perhaps consider cropping and rotating your image such that it shows only the figure. $\endgroup$
    – Roy
    Jul 31, 2014 at 23:16
  • $\begingroup$ Is anyone else bothered that the problem description is not specific about the orientation of the solid? One can freely rotate the solid about $MO$, and for angles close to $0$, the figure will still be accurate... $\endgroup$
    – Hao Ye
    Aug 2, 2014 at 1:16

4 Answers 4

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To find the length of each cube we do the following:

The distance between the vertex $M$ and $O$ is:

$$\sqrt{(0-(-1))^2+(3-3)^2+(0-0)^2}=1$$

Since these two vertices differ from each other by one cube, the length of each cube is $1$.

We suppose that the vertex $N$ has the coordinates $(a,b,c)$.

That means that $N$ lies at the line $x=a$ on the plane $xyz$.

The line $x=a$ on the plane $xy$ differs from the vertex $O$ by two cubes of length $1$.

That means that $a=0+1+1 \Rightarrow a=2$.

Since the vertex $N$ lies on the plane $xz$, the coordinate $y$ should be equal to $0$.

That means that $b=0$.

As for the $z-$ axis, the vertex $N$ is two cubes over $z=0$, that means that $c=2$.

Therefore, the vertex $N$ has coordinates $(2,0,2)$.

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    $\begingroup$ A detailed explanation. Thanks! $\endgroup$
    – Alen
    Aug 1, 2014 at 0:24
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Notice that vertex $N$ lies only in the $xz$-plane. From the coordinates of the other vertices we can see that the side of any square is equivalent to a unit of 1. This means we can just count the blocks to find that the coordinate of vertex $N$ is $(2,0,2)$.

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  • $\begingroup$ Out of all answers I found that yours was the easiest to understand. Thank You! $\endgroup$
    – Alen
    Aug 1, 2014 at 0:24
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If you count the number of dashes along the $xz$ plane, you'll see that coordinate of vertex N is $(2, 0, 2)$.

Note also that each dash (or edge) along the square represent a unit of 1.

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Understand that $N$ is in the $xz$ plane, so y coordinate is 0. Now, $M$ is in the negative $xy$ plane that is why $z=0$. The small cubes has side length of 1 unit. If $Q$ is the point above $M$ such that $MQ$ is the edge of the bigger cuboid then coordinates of Q will be $(-1,3,2)$

Now, the length of $QN$ will be $3\sqrt2$ units with the y coordinate as 0 for N. That gives $(2,0,2)$ as the coordinates.

Alternatively, if try move this point Q so that it lies on the z axis, where y is 0 you will understand, that from thay position, you can walk to N by adding +2 units in the x and z coordinates giving same result.

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