Distance from a point to a plane
length in solid geometry
In Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane, the perpendicular distance to the nearest point on the plane.
It can be found starting with a change of variables that moves the origin to coincide with the given point then finding the point on the shifted plane
a
x
+
b
y
+
c
z
=
d
{\displaystyle ax+by+cz=d}
that is closest to the origin. The resulting point has Cartesian coordinates
(
x
,
y
,
z
)
{\displaystyle (x,y,z)}
:
x
=
a
d
a
2
+
b
2
+
c
2
,
y
=
b
d
a
2
+
b
2
+
c
2
,
z
=
c
d
a
2
+
b
2
+
c
2
{\displaystyle \displaystyle x={\frac {ad}{a^{2}+b^{2}+c^{2}}},\quad \quad \displaystyle y={\frac {bd}{a^{2}+b^{2}+c^{2}}},\quad \quad \displaystyle z={\frac {cd}{a^{2}+b^{2}+c^{2}}}}
.
The distance between the origin and the point
(
x
,
y
,
z
)
{\displaystyle (x,y,z)}
is
x
2
+
y
2
+
z
2
{\displaystyle {\sqrt {x^{2}+y^{2}+z^{2}}}}
.
Converting general problem to distance-from-origin problem
Suppose we wish to find the nearest point on a plane to the point (
X
0
,
Y
0
,
Z
0
{\displaystyle X_{0},Y_{0},Z_{0}}
), where the plane is given by
a
X
+
b
Y
+
c
Z
=
D
{\displaystyle aX+bY+cZ=D}
. We define
x
=
X
−
X
0
{\displaystyle x=X-X_{0}}
,
y
=
Y
−
Y
0
{\displaystyle y=Y-Y_{0}}
,
z
=
Z
−
Z
0
{\displaystyle z=Z-Z_{0}}
, and
d
=
D
−
a
X
0
−
b
Y
0
−
c
Z
0
{\displaystyle d=D-aX_{0}-bY_{0}-cZ_{0}}
, to obtain
a
x
+
b
y
+
c
z
=
d
{\displaystyle ax+by+cz=d}
as the plane expressed in terms of the transformed variables. Now the problem has become one of finding the nearest point on this plane to the origin, and its distance from the origin. The point on the plane in terms of the original coordinates can be found from this point using the above relationships between
x
{\displaystyle x}
and
X
{\displaystyle X}
, between
y
{\displaystyle y}
and
Y
{\displaystyle Y}
, and between
z
{\displaystyle z}
and
Z
{\displaystyle Z}
; the distance in terms of the original coordinates is the same as the distance in terms of the revised coordinates.
Restatement using linear algebra
The formula for the closest point to the origin may be expressed more succinctly using notation from linear algebra. The expression
a
x
+
b
y
+
c
z
{\displaystyle ax+by+cz}
in the definition of a plane is a dot product
(
a
,
b
,
c
)
⋅
(
x
,
y
,
z
)
{\displaystyle (a,b,c)\cdot (x,y,z)}
, and the expression
a
2
+
b
2
+
c
2
{\displaystyle a^{2}+b^{2}+c^{2}}
appearing in the solution is the squared norm
|
(
a
,
b
,
c
)
|
2
{\displaystyle |(a,b,c)|^{2}}
.
The public source identifies “Distance from a point to a plane” as length in solid geometry. This brief keeps that definition visible, then builds a research path around Distance, point and plane.
Why this record matters
A short description can identify a subject without explaining its stakes. For “Distance from a point to a plane”, the useful work is to connect “length in solid geometry” to the records capable of establishing context and consequence.
The citation trail is more important than the brevity of the summary: it shows where individual claims can be examined in context. The source revision retrieved here is dated Oct 21, 2024. The linked authority identifier is Q3030668. None of the 0 selected statements returned an explicit reference.
A concise general-reference account can conceal disagreements about scope, terminology or the weight assigned to individual sources. The source lead contains qualifying language; that uncertainty should survive quotation, summary and reuse. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “Distance from a point to a plane”, its source revision and the description used here.
- Expand the search: follow Distance from a point to a plane primary sources, Distance from a point to a plane archive and Distance research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Distance from a point to a plane”?
- What terminology or title could unlock a more precise catalogue search?
- Which institution is responsible for the underlying evidence?
Search terms from this dossier
This entry incorporates text from “Distance from a point to a plane” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.