MHB How do I determine the coordinates of points on a 3-D coordinate plane?

[ssmith9bmr]

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Can someone please help me with this? I can't for the life of me figure out how to do these points. How do I line up the x, y, and z? I just can't grasp it and can't find anything online.

 

[Evgeny.Makarov]

Concerning problem 1), it is impossible to recover coordinates of the shown point from a two-dimensional picture. That is, there is an infinite number of 3D points that, when drawn on a plane, would fall into the same position. However, we are also given four options, and of those only one corresponds to the required point.

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The most reliable way to find the answer is to draw all four points. To draw a point with coordinates $(x_0,y_0,z_0)$ start with the origin, then move in the direction of the $x$ axis by $x_0$ (in the picture, this means moving bottom left for positive $x_0$), then move in the direction of the $y$ axis by $y_0$ (right for positive $y_0$) and finally in the direction of the $z$ axis by $z_0$ (up for positive $z_0$). But one can also see that the required point has a negative $z$ coordinate, and there is only one corresponding option.

Problem 2)

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Problems 3) and 4) are easier because we have lines connecting the point with the coordinate axes. For 3), we first go up to the $xy$ plane and then left to the $x$ axis. We arrive at $x=4$. Similarly, going up and then towards the $y$ axis we arrive at $y=4$. Finally, going towards the $xy$ plane (where $z=0$) required going up 1 unit, so the $z$ coordinate is $-1$. Note that each time we move along a straight line segment, it must be parallel to one of the axes. The answer to 4) is $(-1,2,-4)$.