# 3-Dimensional Coordinate Geometry Formulas

If you have a feeling that solving the 3-Dimensional Coordinate Geometry problems are difficult. Not anymore, because we have curated the list of 3-Dimensional Coordinate Geometry formulas on this page. Have a glance at the 3D Coordinate Geometry Formulas Sheet and make your computation so simple and quick. Students can find 3D Coordinate Geometry Formulas on the concepts like Distance between two points, Direction ratios of a line, DC’s, Angle between two lines, etc. here below.

## List of 3-Dimensional Coordinate Geometry Formulae

Want to learn and understand the concept of 3-Dimensional Coordinate Geometry thoroughly? Then, take a look at the below-given 3-Dimensional Coordinate Geometry formulas sheet without fail. By using this list of formulas on 3D Coordinate Geometry concepts, you can understand and solve basic to complex Three-Dimensional Coordinate Geometry problems easily and quickly.

1. Distance between two points

If P(x1, y1, z1) and Q(x2, y2, z2) are two points, then distance between them
PQ =

$\sqrt{{\left({x}_{1}-{x}_{2}\right)}^{2}+{\left({y}_{1}-{y}_{2}\right)}^{2}+{\left({z}_{1}-{z}_{2}\right)}^{2}}$

2. Coordinates of division point

Coordinates of the point dividing the line joining two points P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m1 : m2 are
(i) In case of internal division:

$\left(\frac{{m}_{1}{x}_{2}+{m}_{2}{x}_{1}}{{m}_{1}+{m}_{2}},\frac{{m}_{1}{y}_{2}+{m}_{2}{y}_{1}}{{m}_{1}+{m}_{2}},\frac{{m}_{1}{z}_{2}+{m}_{2}{z}_{1}}{{m}_{1}+{m}_{2}}\right)$

(ii) In case of external division:

$\left(\frac{{m}_{1}{x}_{2}-{m}_{2}{x}_{1}}{{m}_{1}-{m}_{2}},\frac{{m}_{1}{y}_{2}-{m}_{2}{y}_{1}}{{m}_{1}-{m}_{2}},\frac{{m}_{1}{z}_{2}-{m}_{2}{z}_{1}}{{m}_{1}-{m}_{2}}\right)$

Note:
(a) Coordinates of the midpoint:

$\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2},\frac{{z}_{1}+{z}_{2}}{2}\right)$

(b) Centroid of a Triangle:

$\left(\frac{{x}_{1}+{x}_{2}+{x}_{3}}{3},\frac{{y}_{1}+{y}_{2}+{y}_{3}}{3},\frac{{z}_{1}+{z}_{2}+{z}_{3}}{3}\right)$

(c) Centroid of a Tetrahedron:
If (xr, yr, zr), r = 1, 2, 3, 4 are vertices of a tetrahedron, then coordinates of its centroid are

$\left(\frac{{\mathrm{x}}_{1}+{\mathrm{x}}_{2}+{\mathrm{x}}_{3}+{\mathrm{x}}_{4}}{4},\frac{{\mathrm{y}}_{1}+{\mathrm{y}}_{2}+{\mathrm{y}}_{3}+{\mathrm{y}}_{4}}{4},\frac{{\mathrm{z}}_{1}+{\mathrm{z}}_{2}+{\mathrm{z}}_{3}+{\mathrm{z}}_{4}}{4}\right)$

3. Direction cosines of a line [Dc’s]

The cosines of the angles made by a line with coordinate axes are called Direction Cosine. If α, β, γ be the angles made by a line with coordinate axes, then direction cosine are l = cos α, m = cos β, n = cos γ and relation between dc’s: l2 + m2 + n2 = 1 i.e. cos2 α + cos2 β + cos2 γ = 1 or sin2 α + sin2 β + sin2 γ = 2

4. Direction ratios of a line

Three numbers which are proportional to the direction cosines of a line are called the direction ratios of that line. If a, b, c are such numbers then
a, b, c dr s ⇔

$\frac{a}{\ell }=\frac{b}{m}=\frac{c}{n}$

Further we may observe that in above case

$\frac{\ell }{a}=\frac{m}{b}=\frac{n}{c}$

= ±

$\frac{\sqrt{{\ell }^{2}+{m}^{2}+{n}^{2}}}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}=±\frac{1}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}$

⇒ l = ±

$\frac{a}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}$

, m = ±

$\frac{b}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}$

, n = ±

$\frac{c}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}$

5. Direction cosines of a line joining two points

Let P = (x1, y1, z1) and Q = (x2, y2, z2); then
(i) dr’s of PQ: (x2 – x1), (y2 – y1), (z2 – z1)

(ii) dc’s of PQ:

$\frac{{x}_{2}-{x}_{1}}{PQ},\frac{{y}_{2}-{y}_{1}}{PQ},\frac{{z}_{2}-{z}_{1}}{PQ}$

i.e.

$\frac{{x}_{2}-{x}_{1}}{\sqrt{\mathrm{\Sigma }{\left({x}_{2}-{x}_{1}\right)}^{2}}},\frac{{y}_{2}-{y}_{1}}{\sqrt{\mathrm{\Sigma }{\left({x}_{2}-{x}_{1}\right)}^{2}}},\frac{{z}_{2}-{z}_{1}}{\sqrt{\mathrm{\Sigma }{\left({x}_{2}-{x}_{1}\right)}^{2}}}$

6. Angle between two lines

(i) When direction cosines of the lines are given:
If l1, m1, n1 and l2, m2, n2 are dc’s of given two lines, then the angle θ between them is given by
* cos θ = l1l2 + m1m2 + n1n2
* sin θ =

$\sqrt{{\left({\ell }_{1}{m}_{2}-{\ell }_{2}{m}_{1}\right)}^{2}+{\left({m}_{1}{n}_{2}-{m}_{2}{n}_{1}\right)}^{2}+{\left({n}_{1}{\ell }_{2}-{n}_{2}{\ell }_{1}\right)}^{2}}$

(ii) When direction ratios of the lines are given:
If a1, b1, c1 and a2, b2, c2 are dr’s of given two lines, then the angle θ between them is given by
* cos θ =

$\frac{{a}_{1}{a}_{2}+{b}_{1}{b}_{2}+{c}_{1}{c}_{2}}{\sqrt{{a}_{1}^{2}+{b}_{1}^{2}+{c}_{1}^{2}}\sqrt{{a}_{2}^{2}+{b}_{2}^{2}+{c}_{2}^{2}}}$

* sin θ =

$\frac{\sqrt{\sum {\left(a,{b}_{2}-{a}_{2}{b}_{1}\right)}^{2}}}{\sqrt{{a}_{1}^{2}+{b}_{1}^{2}+{c}_{1}^{2}}\sqrt{{a}_{2}^{2}+{b}_{2}^{2}+{c}_{2}^{2}}}$

7. Conditions of parallelism and perpendicularity of two lines

(i) When Dc’s of two lines AB and CD say l1, m1, n1 and l2, m2, n2 are known, then
AB || CD ⇔ l1 = l2, m1 = m2, n1 = n2
AB ⊥ CD ⇔ l1l2 + m1m2 + n1n2 = 0

(ii) When dr’s of two lines AB & CD, say a1; bj, Cj and a2, b2, c2 are known, then
AB || CD ⇔

$\frac{{a}_{1}}{{a}_{2}}=\frac{{b}_{1}}{{b}_{2}}=\frac{{c}_{1}}{{c}_{2}}$

AB ⊥ CD ⇔ a1a2 + b1b2 + c1c2 = 0

8. Projection of a line segment joining two points on a line

(i) Let PQ be a line segment where P = (x1, y1, z1) and Q = (x2, y2, z2); and AB be a given line with dc’s as l, m, n. Then projection of PQ is P’Q’ = l(x2 – x1) + m (y2 – y1 + n (z2 – z1)

(ii) If a, b, c are the projections of a line segment on coordinate axes, then length of the segment =

$\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}$

(iii) If a, b, c are projections of a line segment on coordinate axes then its dc’s are

$±\frac{a}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}},±\frac{b}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}},±\frac{c}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}$

9. Cartesian equation of a line passing through a given point & given direction ratios

Cartesian equation of a straight line passing through a fixed point (x1, y1, z1) and having direction ratios a, b, c is

$\frac{x-{x}_{1}}{a}=\frac{y-{y}_{1}}{b}=\frac{z-{z}_{1}}{c}$

10. Cartesian equation of a line passing through two given points

The cartesian equation of a line passing through two given points (x1, y1, z1) and Q = (x2, y2, z2) is given by

$\frac{x-{x}_{1}}{{x}_{2}-{x}_{1}}=\frac{y-{y}_{1}}{{y}_{2}-{y}_{1}}=\frac{z-{z}_{1}}{{z}_{2}-{z}_{1}}$

11. Perpendicular distance of a point from a line

(a) Cartesian Form:
To find the perpendicular distance of a given point (α, β, γ) from a given line

$\frac{x-{x}_{1}}{a}=\frac{y-{y}_{1}}{b}=\frac{z-{z}_{1}}{c}$

$\frac{x-{x}_{1}}{a}=\frac{y-{y}_{1}}{b}=\frac{y-{z}_{1}}{c}$

L(x1 + aλ, y1 + bλ, z1 + cλ)
Let L be the foot of the perpendicular drawn from P (α, β, γ) on the line

$\frac{x-{x}_{1}}{a}=\frac{y-{y}_{1}}{b}=\frac{z-{z}_{1}}{c}$

Let the coordinates of L be (x1 + aλ, y1 + bλ, + z1 + cλ). Then direction ratios of PL are x1 + aλ – α, y1 + bλ – β, z1 + cλ – γ.
Direction ratio of AB are a, b, c. Since PL is perpendicular to AB, therefore
(x1 + aλ – α) a + (y1 + bλ – β) b + (z1 + cλ – γ) c = 0 =
⇒ λ =

$\frac{a\left(\alpha -{x}_{1}\right)+b\left(\beta -{y}_{1}\right)+c\left(\gamma -{z}_{1}\right)}{{a}^{2}+{b}^{2}+{c}^{2}}$

Puting this value of λ in (x1 + aλ, y1 + bλ, z1 + cλ), we obtain coordinates of L. Now, using distance formula we can obtain the length PL.
* Using “L” as mid point, we can find image of P with respect to given plane

(b) Co-ordinate’s of “L” let they are (p, q, r)

$\frac{p-\alpha }{a}=\frac{q-\beta }{b}=\frac{r-\gamma }{c}=\frac{-\left(a\alpha +b\beta +c\gamma +d\right)}{{a}^{2}+{b}^{2}+{c}^{2}}$

where “L” is foot of perpendicular again image of (α, β, γ) is (r, s, t) then

$\frac{r-\alpha }{a}=\frac{s-\beta }{b}=\frac{t-\gamma }{c}=\frac{-2\left(a\alpha +b\beta +c\gamma +d\right)}{\left({a}^{2}+{b}^{2}+{c}^{2}\right)}$

12. Plane

(i) General equation of a plane: ax + by + cz + d = 0

(ii) Equation of a plane passing through a given point:
The general equation of a plane passing through a point (x1, y1, z1) is a(x – x1) + b(y – y1) + c(z – z1) = 0, where a, b and c are constants.

(iii) Intercept form of a plane:
The equation of a plane intercepting lengths a, b and c with x-axis , y-axis and z-axis respectively is

$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$

(iv) Normal Form:
If l, m, n are direction cosines of the normal to a given plane which is at a distance p from the origin, then the equation of the plane is lx + my + nz = p.

(v) The reflection of the plane ax + by + cz + d = 0 on the plane, a1x + b1y + c1z + d1 = 0 is
2(aa1 + bb1 + cc1) (a1 + b1y + c1z + d1)
= (a12 + a22 + a32) × (ax + by + cz + d)

13. Angle between two planes in Cartesian form

The angle θ between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is given by
cos θ =

$\frac{{a}_{1}{a}_{2}+{b}_{1}{b}_{2}+{c}_{1}{c}_{2}}{\sqrt{{a}_{1}^{2}+{b}_{1}^{2}+{c}_{1}^{2}}\sqrt{{a}_{2}^{2}+{b}_{2}^{2}+{c}_{2}^{2}}}$

14. Distance of a point from a plane

The length of the perpendicular from a point P(x1, y1, z1) to the plane ax + by + cz + d = 0 is given by

$\frac{|a{x}_{1}+b{y}_{1}+c{z}_{1}+d|}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}$

15. Equation of plane bisecting the angle between two given planes

The equation of the planes bisecting the angles between the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 are

$\frac{\left({a}_{1}x+{b}_{1}y+{c}_{1}z+{d}_{1}\right)}{\sqrt{{a}_{1}^{2}+{b}_{1}^{2}+{c}_{1}^{2}}}=±\frac{\left({a}_{2}x+{b}_{2}y+{c}_{2}z+{d}_{2}\right)}{\sqrt{{a}_{2}^{2}+{b}_{2}^{2}+{c}_{2}^{2}}}$

If d1 and d2 are +ve then following table can be utilized to write acute angle bisector or obtuse angle bisector.

16. Condition of coplanarity of two lines

If the line

are coplanar, then

$|\begin{array}{ccc}{x}_{2}-{x}_{1}& {y}_{2}-{y}_{1}& {z}_{2}-{z}_{1}\\ {\ell }_{1}& {m}_{1}& {n}_{1}\\ {\ell }_{2}& {m}_{2}& {n}_{2}\end{array}|$

= 0

Area of a triangle:
If Ayz, Azx, Axy be the projection of an area A on the coordinate plane yz, zx and xy respectively then A =

$\sqrt{{A}_{yz}^{2}+{A}_{zx}^{2}+{A}_{xy}^{2}}$

If the vertices of a triangle are (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) then

${A}_{xy}=\frac{1}{2}|\begin{array}{lll}{x}_{1}& {y}_{1}& 1\\ {x}_{2}& {y}_{2}& 1\\ {x}_{3}& {y}_{3}& 1\end{array}|,{A}_{yz}=\frac{1}{2}|\begin{array}{lll}{y}_{1}& {z}_{1}& 1\\ {y}_{2}& {z}_{2}& 1\\ {y}_{3}& {z}_{3}& 1\end{array}|,{A}_{zx}=\frac{1}{2}|\begin{array}{ccc}{z}_{1}& {x}_{1}& 1\\ {z}_{2}& {x}_{2}& 1\\ {z}_{3}& {x}_{3}& 1\end{array}|$

17. Sphere

(i) The equation of a sphere with centre (a, b, c) and radius R is (x – a)2 + (y – b)2 + (z – c)2 = R2

(ii) General equation of a sphere:
The equation x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 represents a sphere with centre (- u, – v, – w) and radius =

$\sqrt{{u}^{2}+{v}^{2}+{w}^{2}-d}$

(iii) Diameter form of the equation of a sphere:
If (x1, y1, z1) and (x2, y2, z2) are the coordinates of the extremities of a diameter of a sphere, then its equation is
(x – x1) (x – x2) + (y – y1) (y – y2) + (z – z1) (z – z2) = 0.
An equation of the form
ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy = 0 is a homogeneous equation of 2nd degree may represent pair of planes if

$|\begin{array}{lll}a& h& g\\ h& b& f\\ g& f& c\end{array}|$

= 0 and angle between two planes is
θ = tan-1 =

$\left[\frac{2\sqrt{{f}^{2}+{g}^{2}+{h}^{2}-bc-ca-ab}}{|a+b+c|}\right]$

Let a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 be the equation of any two planes, taken together then
a1x + b1y + c1z + d1 = 0 = a2x + b2y + c2z + d2
Note:

• x2 + y2 + z2 = r2 is equation of sphere with centre (0, 0, 0) and radius “r”
• Equation of sphere will have coeff. of x2, y2, z2 equal and coeff. of term xy, yz, zx must be absent.
• Equation of tangent plane at any point (x1, y1, z1) of the sphere x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 is
xx1 + yy1 + zz1 + u(x + x1) + v(y + y1) + w(z + z1) + d = 0
• The plane lx + my + nz = p will touch the sphere x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 if (ul + vm + wn + p)2 = (l2 + m2 + n2) (u2 + v2 + w2 – d)

18. Distance between the parallel planes :

Let two parallel planes are a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0
and distance between plane

$=\frac{|{d}_{1}-{d}_{2}|}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}$

19. Coplanarity of lines:

(a) Let lines are

coplanar then

$|\begin{array}{ccc}{x}_{2}-{x}_{1}& {y}_{2}-{y}_{1}& {z}_{2}-{z}_{1}\\ {\ell }_{1}& {m}_{1}& {n}_{1}\\ {\ell }_{2}& {m}_{2}& {n}_{2}\end{array}|$

= 0
and

$|\begin{array}{ccc}x-{x}_{1}& y-{y}_{1}& z-{z}_{1}\\ {\ell }_{1}& {m}_{1}& {n}_{1}\\ {\ell }_{2}& {m}_{2}& {n}_{2}\end{array}|$

= 0

(b) Let lines are

$\frac{x-{x}_{1}}{\ell }=\frac{y-{y}_{1}}{m}=\frac{z-{z}_{1}}{n}$

and a1x + b1y + c1z = d1 = 0 = a2x + b2y + c2z + d2 the condition for coplanarity is

$\frac{{a}_{1}{x}_{1}+{b}_{1}{y}_{1}+{c}_{1}{z}_{1}+{d}_{1}}{{a}_{2}{x}_{1}+{b}_{2}{y}_{1}+{c}_{2}{z}_{1}+{d}_{2}}=\frac{{a}_{1}\ell +{b}_{1}m+{c}_{1}n}{{a}_{2}\ell +{b}_{2}m+{c}_{2}n}$

(c) Let lines are a1 + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 a3x + b3y + c3z + d3 = 0 = a4x + b4y + c4z + d4 then condition that lines are coplanar is
Δ =

$|\begin{array}{llll}{a}_{1}& {b}_{1}& {c}_{1}& {d}_{1}\\ {a}_{2}& {b}_{2}& {c}_{2}& {d}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}& {d}_{3}\\ {a}_{4}& {b}_{4}& {c}_{4}& {d}_{4}\end{array}|$

= 0

20. Family of planes:

Equation of a plane passing through line of intersection of the planes ax + by + cz + d = 0 and a1x + b1y + c1z + d1 = 0 can be represented by the equation (ax + by + cz + d) + λ(a1x + b1y + c1z + d1) = 0

21. Skew lines:

Two straight lines in space are called skew lines if neither j they are parallel nor they intersect each other the distance between j then may be calculated by using following formulae

Let S.D. lie along the line

$\frac{x-\alpha }{\ell }=\frac{y-\beta }{m}=\frac{z-\gamma }{n}$

∴ S.D. = |l(x2 – x1) + m (y2 – y1) + n (z2 – z1)|
and the equation of shortest distance is

22. Volume of tetrahedron:

If the vertices of tetrahedron are (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) and (x4, y4, z4) then
Volume =

$=\frac{1}{6}|\begin{array}{llll}{x}_{1}& {y}_{1}& {z}_{1}& 1\\ {x}_{2}& {y}_{2}& {z}_{2}& 1\\ {x}_{3}& {y}_{3}& {z}_{3}& 1\\ {x}_{4}& {y}_{4}& {z}_{4}& 1\end{array}|$

= 0