Coordinates of a point in space

Drop a perpendicular from a point $P$ to each of the three coordinate planes. The signed length of the perpendicular to the $YZ$-plane is called the $x$-coordinate of $P$, the perpendicular to the $ZX$-plane gives the $y$-coordinate, and the perpendicular to the $XY$-plane gives the $z$-coordinate. Collected together they form the ordered triple

where: $x$ is the distance of $P$ from the $YZ$-plane (measured along a line parallel to the $x$-axis), $y$ is its distance from the $ZX$-plane, and $z$ is its distance from the $XY$-plane; a positive value means the perpendicular falls on the positive side of that axis, a negative value the opposite side. The same triple can be reached by construction instead of measurement: start at the origin $O$, move a distance $x$ along the $x$-axis to a point $A$, then a distance $y$ parallel to the $y$-axis to a point $B$, then a distance $z$ parallel to the $z$-axis to land on $P$. Both routes — perpendiculars in, or moves out — describe the same one-to-one correspondence between points of space and ordered triples of real numbers. Four facts fall straight out of this correspondence. The origin is $(0,0,0)$, the unique point at zero distance from all three planes. A point lies on an axis exactly when two of its three coordinates are zero: on the $x$-axis it has the form $(x,0,0)$, on the $y$-axis $(0,y,0)$, on the $z$-axis $(0,0,z)$. A point lies on a coordinate plane exactly when one coordinate is zero: $(x,y,0)$ on the $XY$-plane, $(0,y,z)$ on the $YZ$-plane, $(x,0,z)$ on the $ZX$-plane. The sign pattern of $(x,y,z)$ — which of the three coordinates are positive and which negative — sorts every point into one of eight octants, the 3-D analogue of the four quadrants of the plane. All three positive is the first octant; the remaining seven combinations of signs give the other seven, conventionally labelled but rarely named individually at this level. Two triples represent the same point only if all three coordinates agree: $(x_1,y_1,z_1)=(x_2,y_2,z_2)$ requires $x_1=x_2$ and $y_1=y_2$ and $z_1=z_2$ simultaneously — matching just one or two coordinates is not enough.

Question: Plot $P(2,3,4)$ by describing the path from the origin, state its octant, and find the coordinates of its mirror image in the $YZ$-plane and in the $y$-axis.

Step 1. From the origin $O(0,0,0)$, move $2$ units along the positive $x$-axis to $A(2,0,0)$, then $3$ units parallel to the positive $y$-axis to $B(2,3,0)$ — note $B$ lies on the $XY$-plane since its $z$-coordinate is $0$ — then $4$ units parallel to the positive $z$-axis to reach $P(2,3,4)$.

Step 2. Since $x=2$, $y=3$, $z=4$ are all positive, $P$ lies in the first octant.

Step 3. Reflection in the $YZ$-plane flips only the $x$-coordinate (Step 2 of the derivation): the image is $(-2,3,4)$.

Step 4. Reflection in the $y$-axis flips the two coordinates perpendicular to that axis, $x$ and $z$ (Step 4 of the derivation): the image is $(-2,3,-4)$. Final answer: $P(2,3,4)$ is in the first octant; its mirror image in the $YZ$-plane is $(-2,3,4)$, and its mirror image in the $y$-axis is $(-2,3,-4)$.

Competitive stretch

For $Q(5,-2,7)$, the derivation's plane rule gives its image in the $XY$-plane as $(5,-2,-7)$ (only $z$ flips), and the axis rule gives its image in the $x$-axis as $(5,2,-7)$ (both $y$ and $z$ flip) — matching the two cases worked directly from the definition, confirming the general rules read off the geometry correctly.

• Writing coordinates in the wrong order: $(2,3,4)$ and $(3,2,4)$ are different points, because $(x,y,z)$ is an ordered triple, exactly as in 2-D coordinate geometry.
• Thinking a point "on an axis" needs only one coordinate zero — it needs two, since an axis is where two coordinate planes intersect.
• Confusing reflection in a plane (one sign flips) with reflection in an axis (two signs flip) — the derivation's Step 2 versus Step 4 is the rule to lean on.
• Assuming every problem lives in the friendly all-positive first octant; negative coordinates are perfectly valid and just mean the point is on the negative side of that axis.

Ask the tutor: if octant identification or plane-versus-axis reflections still feel shaky, ask the AI tutor for a few more points to classify and reflect.