Coordinates of a Point in Space

Fig 11.2 is the foundational picture for how a point gets its three coordinates. It shows a single point $P$ floating in the first octant (where all coordinates are positive), and the figure traces the path you would take to measure its $x$, $y$, and $z$ values.

The three axes are drawn from the origin $O$: the $z$-axis points straight up, the $y$-axis points to the right, and the $x$-axis points down and to the left (so that the three axes are not all in the same plane). The point $P(x,y,z)$ sits in the upper-right region of the diagram. Directly below it, on the horizontal $XY$-plane, is the foot $M(x,y,0)$. The vertical segment $MP$ is drawn dashed and labelled $z$ — it is the perpendicular from $P$ down to the $XY$-plane.

On the $XY$-plane itself, the figure shows a dashed rectangle (actually a parallelogram in perspective) with vertices $O$, $L$, $M$, and $C$. $L$ lies on the $x$-axis, and $C$ lies on the $y$-axis. The segment $OL$ is labelled $x$, and the segment $LM$ (which runs parallel to the $y$-axis) is labelled $y$. So the path from $O$ to $P$ is broken into three perpendicular steps: go $x$ units along the $x$-axis to $L$, then $y$ units parallel to the $y$-axis to $M$, then $z$ units straight up to $P$.

The physical idea is simple but crucial: the three coordinates of a point are just three perpendicular distances measured from the coordinate planes. The $x$-coordinate is the perpendicular distance from the $YZ$-plane, the $y$-coordinate is the distance from the $ZX$-plane, and the $z$-coordinate is the distance from the $XY$-plane. The figure makes this concrete by showing the perpendicular drops one at a time.

Here $OL$ is the distance from the origin to the point $L$ on the $x$-axis, $LM$ is the distance from $L$ to $M$ measured parallel to the $y$-axis, and $MP$ is the vertical distance from $M$ up to $P$. These three numbers, taken in order, uniquely identify the point $P$ in space.

The textbook uses this figure to establish the one-to-one correspondence between points in space and ordered triples $(x,y,z)$. Given any point $P$, you can find its coordinates by dropping perpendiculars as shown. Conversely, given any triple $(x,y,z)$, you can locate $P$ by first fixing $L$ on the $x$-axis at distance $x$, then moving parallel to the $y$-axis to reach $M$ in the $XY$-plane, then rising vertically by $z$. …

What Fig. 11.3 Shows

The figure presents an alternative way to visualise the coordinates of a point in space — one that avoids dropping perpendiculars onto planes. Instead, it builds a rectangular box (a rectangular parallelepiped) whose three edges lie along the coordinate axes, meeting at the origin O.

Three points are marked on the axes: A on the x‑axis, B on the y‑axis, and C on the z‑axis. The distances from the origin to these points are labelled $x$, $y$, and $z$ respectively — so $OA = x$, $OB = y$, $OC = z$. The three edges from O to A, O to B, and O to C are drawn as dashed lines, indicating they are the hidden back edges of the box.

From A, B, and C, planes are drawn parallel to the coordinate planes: through A a plane parallel to the YZ‑plane, through B a plane parallel to the ZX‑plane, and through C a plane parallel to the XY‑plane. These three planes intersect at the point P, which is the far corner of the box opposite O. P is highlighted in indigo. The remaining three corners of the box — D, E, and F — are shown as solid points. The top face and two side faces of the box are shaded light blue to give a sense of depth.

The Physical Idea

The core insight is that the coordinates $(x, y, z)$ of a point P are simply the distances from the origin to the points where the three coordinate planes, shifted to pass through P, meet the axes. Instead of measuring perpendicular distances to planes directly, you can think of constructing a box whose three dimensions are exactly $x$, $y$, and $z$.

This is the "alternatively" method described in the textbook: given a point P, draw three planes through P that are parallel to the coordinate planes. Where these planes cut the axes gives you the three numbers $x$, $y$, $z$. Conversely, given the numbers $x$, $y$, $z$, you locate A, B, C on the axes, draw the three planes through them parallel to the coordinate planes, and their intersection is P.

Key Formula Developed with This Figure

The figure directly leads to the coordinate representation of any point in space:

Here:

• $O$ is the origin $(0,0,0)$
• $A$ is the point on the x‑axis such that $OA = x$
• $B$ is the point on the y‑axis such that $OB = y$
• $C$ is the point on the z‑axis such that $OC = z$
• $P$ is the corner of the rectangular box opposite $O$, obtained by drawing planes through $A$, $B$, $C$ parallel to the coordinate planes …