Q.Three vertices of a parallelogram ABCD are A(3,−1,2), B(1,2,−4) and C(−1,1,2). Find the coordinates of the fourth vertex.
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Concept understanding — 3D Coordinate Geometry
3D Coordinate Geometry
You already know 2D coordinate geometry — the xy-plane where every point is described by two numbers (x,y). Now imagine lifting that plane into the air. That is three-dimensional geometry.
The Intuition: Three Numbers, One Point
In the real world you rarely locate something with just two numbers. To describe where a book sits on a shelf you might say: "third shelf up, fourth book from the left, and it is the one nearest the wall." That is three pieces of information — height, sideways position, and depth.
In 3D coordinate geometry we do exactly this. We keep the familiar x and y axes (which define a flat floor) and add a third axis — the z-axis — pointing straight up. Every point in space now needs three numbers: (x,y,z).
Note
The three axes are mutually perpendicular. Picture the corner of a room: two floor edges give the x- and y-axes, and the vertical edge where the walls meet gives the z-axis.
The Precise Statement
Definition: A rectangular 3D coordinate system consists of three mutually perpendicular number lines — the x-axis, y-axis and z-axis — meeting at a common point, the originO(0,0,0). Any point P in space is uniquely represented by an ordered triple (x,y,z), where:
x = signed distance from the yz-plane,
y = signed distance from the zx-plane,
z = signed distance from the xy-plane.
P=(x,y,z)
How to Read a 3D Point
Take the point A(2,−3,4). Start at the origin. Move 2 units along the x-axis. From there move −3 units parallel to the y-axis (backward, because it is negative). From that spot move 4 units parallel to the z-axis (upward). You have reached A.
Watch out
The order matters absolutely. (2,−3,4) is not the same point as (2,4,−3). Always follow the sequence: x first, then y, then z.
The Three Coordinate Planes
Each pair of axes determines a plane:
Plane
Equation
Description
xy-plane
z=0
the floor — all points with zero height
yz-plane
x=0
one wall — all points with zero x
zx-plane
y=0
the other wall — all points with zero y
These three planes cut space into 8 octants (the 3D analogue of the four quadrants of the plane). The first octant is where x>0, y>0 and z>0.
Distance Between Two Points
This is the natural extension of the 2D distance formula. For P(x1,y1,z1) and Q(x2,y2,z2):
PQ=(x2−x1)2+(y2−y1)2+(z2−z1)2
It is just the diagonal of a rectangular box whose edges are the differences in x, y and z. The distance of P from the origin is the special case OP=x12+y12+z12.
Section Formula (Internal Division)
If R divides the segment joining P(x1,y1,z1) and Q(x2,y2,z2) internally in the ratio m:n, then:
R=(m+nmx2+nx1,m+nmy2+ny1,m+nmz2+nz1)
This is the same pattern as the 2D section formula, applied to three coordinates instead of two.
Tip
For the midpoint, set m=n=1:
M=(2x1+x2,2y1+y2,2z1+z2)
What Comes Next
Once you are comfortable with points, distances and section division, the natural next steps (in later study) are direction cosines and the equations of lines and planes in space. For now, remember the core idea: 3D coordinate geometry is 2D geometry with one extra dimension — every formula you already know simply gains a third term.
3D Coordinate Geometry is the heart of the NCERT Class 11 Mathematics chapter Introduction to Three Dimensional Geometry, matching searches such as "3D coordinate geometry formulas class 11 maths" or "distance and section formula in 3D important questions". The octants, coordinate planes, distance formula and section formula introduced here carry real weightage in CBSE Class 11 exams and form the groundwork for the Class 12 three-dimensional geometry of lines and planes, as well as the coordinate-geometry sections of JEE Main and state CETs.
Concept: Diagonals of a parallelogram bisect each other
In a parallelogram, the diagonals intersect at their mutual midpoint. Let the fourth vertex be D(x,y,z).
The diagonals are AC and BD. Their midpoint must coincide:
Midpoint of AC=(23+(−1),2−1+1,22+2)=(1,0,2)
Midpoint of BD=(21+x,22+y,2−4+z)
Equating the two midpoints:
21+x=1⟹x=1
22+y=0⟹y=−2
2−4+z=2⟹z=8
✓Final answer
The fourth vertex is D(1,−2,8).
In a parallelogram, diagonals bisect each other, so the midpoint of AC equals the midpoint of BD. Using this condition, the fourth vertex is D(1,−2,8).
The defining property of a parallelogram is that its diagonals bisect each other. This means the point where the diagonals cross is the midpoint of both diagonals. If we know three vertices A, B, and C, we can find the fourth vertex D by equating the midpoint of diagonal AC with the midpoint of diagonal BD.
Why does this work? Because the diagonals of a parallelogram always meet at their mutual midpoint, this single condition captures the entire geometry of the figure. Once we enforce that the diagonals share a common midpoint, the fourth vertex is uniquely determined.
Let's denote the unknown fourth vertex as D(x,y,z).
This gives us three equations (one for each coordinate):
21+x=1,22+y=0,2−4+z=2
Solve for x, y, and z.
From the first equation:
21+x=1⟹1+x=2⟹x=1
From the second equation:
22+y=0⟹2+y=0⟹y=−2
From the third equation:
2−4+z=2⟹−4+z=4⟹z=8
Tip
You can verify your answer by checking that AB=DC (opposite sides are parallel and equal). Here, AB=(−2,3,−6) and DC=(2,−3,6)=−AB, which confirms the parallelogram property when we account for direction.
✓Final answer
The coordinates of the fourth vertex are D(1,−2,8).
Q.Write True/False: The distance between points (1,0,0) and (0,1,0) is 2.
›Reveal solutionSolution
Using the 3D distance formula, the distance between (1,0,0) and (0,1,0) works out to exactly 2, confirming the statement.
Distance between (x1,y1,z1) and (x2,y2,z2) is d=(x2−x1)2+(y2−y1)2+(z2−z1)2.
Here (x1,y1,z1)=(1,0,0) and (x2,y2,z2)=(0,1,0):
d=(0−1)2+(1−0)2+(0−0)2=1+1+0=2.
✓Final answer
True.
CBSE 2026Set ANNUAL1 mark
Q.Find distance between the points (4,0,0) and (−4,0,0).
›Reveal solutionSolution
Distance formula in 3D gives d=(x2−x1)2+(y2−y1)2+(z2−z1)2; here it simplifies to 8, the direct separation along the x-axis.
Points: (4,0,0) and (−4,0,0).
d=(−4−4)2+(0−0)2+(0−0)2=(−8)2=64=8.
✓Final answer
Distance =8 units.
CBSE 2026Set 1A1 markMCQ
Q.XOZ-plane divides the line joining the points (2,3,1) and (6,7,1) in the ratio -
(1) 3:7
(2) 2:7
(3) −3:7
(4) −2:4
›Reveal solutionSolution
Setting the y-coordinate to 0 in the section formula gives ratio −3:7.
Let the XOZ-plane (equation y=0) divide the join of (2,3,1) and (6,7,1) in the ratio k:1. The y-coordinate of the division point is
y=k+17k+3=0⇒7k+3=0⇒k=−73.
So the ratio k:1=−3:7 (the negative sign shows external division).
✓Final answer
The ratio is −3:7, i.e. option (3).
CBSE 2025Set ANNUAL1 markMCQ
Q.The coordinates of origin in three dimensional geometry are
(a) (0,0)
(b) (0,0,0)
(c) (0,0,0,0)
(d) (0,0,0,0,0)
›Reveal solutionSolution
The origin in 3D geometry is (0,0,0).
In 3D space, every point needs three coordinates (along the x, y, z axes) to be located. The origin O is the common point where all three coordinate axes intersect, so all three coordinates are zero.
✓Final answer
(b) (0,0,0).
CBSE 2025Set ANNUAL1 markMCQ
Q.The distance between the points (2,3,5) and (4,3,1) is
Q.Fill in the blank: A point with co-ordinates (2,3,0) lies in the ____ plane.
›Reveal solutionSolution
Any point of the form (x,y,0), with z-coordinate zero, lies in the xy-plane.
In three-dimensional geometry, the xy-plane consists of all points whose z-coordinate is 0; the yz-plane consists of points whose x-coordinate is 0; the xz-plane consists of points whose y-coordinate is 0.
The point (2,3,0) has z=0, so it lies in the xy-plane.
✓Final answer
The point (2,3,0) lies in the xy-plane.
CBSE 2025Set ANNUAL1 markMCQ
Q.Match the column: Column A entry 'Distance of the point (2,4,5) from the xz plane' — find the matching value from Column B.
(a) 2
(b) 8
(c) 32
(d) 1−tan2x2tanx
(e) sin2x
(f) 10
(g) 20
(h) 1+tan2x2tanx
(i) 4
›Reveal solutionSolution
The distance of any point (x,y,z) from the xz-plane equals ∣y∣; here y=4.
The xz-plane consists of all points with y=0. The perpendicular distance of a point (x,y,z) from the xz-plane is ∣y∣.
For the point (2,4,5), the distance from the xz-plane is ∣4∣=4.
✓Final answer
The distance of (2,4,5) from the xz plane matches option (i) 4.
CBSE 2025Set ANNUAL1 mark
Q.A point lies in the xz plane. What will be its y-coordinate?
›Reveal solutionSolution
The xz-plane is defined as the set of all points with y=0.
In 3D coordinate geometry, the xz-plane consists of all points of the form (x,0,z) — i.e. every point on this plane has y-coordinate equal to 0.
✓Final answer
A point lying in the xz plane has y-coordinate 0.
CBSE 2024Set ANNUAL1 markMCQ
Q.The distance of the point (1,−3,4) from x-axis is
(a) 5
(b) 1
(c) 26
(d) None of these
›Reveal solutionSolution
The distance from a point to the x-axis depends only on its y and z coordinates (the x-coordinate doesn't matter, since the foot of the perpendicular slides along the axis).
For a point (x,y,z), its perpendicular distance from the x-axis is:
d=y2+z2
(This is because the foot of the perpendicular from the point to the x-axis is (x,0,0), and the distance formula between (x,y,z) and (x,0,0) simplifies to y2+z2.)