3-Dimensional Coordinate Geometry Formulas

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 (x_r, y_r, z_r), 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
\(\frac{\ell}{a}=\frac{m}{b}=\frac{n}{c}\) = ± \(\frac{\sqrt{\ell^{2}+m^{2}+n^{2}}}{\sqrt{a^{2}+b^{2}+c^{2}}}=\pm \frac{1}{\sqrt{a^{2}+b^{2}+c^{2}}}\)
⇒ 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: \(\frac{x_{2}-x_{1}}{P Q}, \frac{y_{2}-y_{1}}{P Q}, \frac{z_{2}-z_{1}}{P Q}\)
i.e. \(\frac{x_{2}-x_{1}}{\sqrt{\Sigma\left(x_{2}-x_{1}\right)^{2}}}, \frac{y_{2}-y_{1}}{\sqrt{\Sigma\left(x_{2}-x_{1}\right)^{2}}}, \frac{z_{2}-z_{1}}{\sqrt{\Sigma\left(x_{2}-x_{1}\right)^{2}}}\)

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
\(\pm \frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}, \pm \frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}, \pm \frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}\)

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)
\(\frac{p-\alpha}{a}=\frac{q-\beta}{b}=\frac{r-\gamma}{c}=\frac{-(a \alpha+b \beta+c \gamma+d)}{a^{2}+b^{2}+c^{2}}\)
where “L” is foot of perpendicular again image of (α, β, γ) is (r, s, t) then
\(\frac{r-\alpha}{a}=\frac{s-\beta}{b}=\frac{t-\gamma}{c}=\frac{-2(a \alpha+b \beta+c \gamma+d)}{\left(a^{2}+b^{2}+c^{2}\right)}\)

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 \(\frac{\left|a x_{1}+b y_{1}+c z_{1}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}\)

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
\(\frac{\left(a_{1} x+b_{1} y+c_{1} z+d_{1}\right)}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}=\pm \frac{\left(a_{2} x+b_{2} y+c_{2} z+d_{2}\right)}{\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\)
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 \(\frac{x-x_{1}}{\ell_{1}}=\frac{y-y_{1}}{m_{1}}=\frac{z-z_{1}}{n_{1}} \text { and } \frac{x-x_{2}}{\ell_{2}}=\frac{y-y_{2}}{m_{2}}=\frac{z-z_{2}}{n_{2}}\) are coplanar, then
\(\left|\begin{array}{ccc}x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1} \\\ell_{1} & m_{1} & n_{1} \\\ell_{2} & m_{2} & n_{2}\end{array}\right|\) = 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 = \(\sqrt{A_{y z}^{2}+A_{z x}^{2}+A_{x y}^{2}}\)
If the vertices of a triangle are (x₁, y₁, z₁), (x₂, y₂, z₂) and (x₃, y₃, z₃) then
\(A_{x y}=\frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\x_{2} & y_{2} & 1 \\x_{3} & y_{3} & 1\end{array}\right|, A_{y z}=\frac{1}{2}\left|\begin{array}{lll}y_{1} & z_{1} & 1 \\y_{2} & z_{2} & 1 \\y_{3} & z_{3} & 1\end{array}\right|, A_{z x}=\frac{1}{2}\left|\begin{array}{ccc}z_{1} & x_{1} & 1 \\z_{2} & x_{2} & 1 \\z_{3} & x_{3} & 1\end{array}\right|\)

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 2^nd degree may represent pair of planes if
\(\left|\begin{array}{lll}a & h & g \\h & b & f \\g & f & c\end{array}\right|\) = 0 and angle between two planes is
θ = tan^-1 = \(\left[\frac{2 \sqrt{f^{2}+g^{2}+h^{2}-b c-c a-a b}}{|a+b+c|}\right]\)
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 \(=\frac{\left|d_{1}-d_{2}\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}\)

19. Coplanarity of lines:

(a) Let lines are
\(\frac{x-x_{1}}{\ell_{1}}=\frac{y-y_{1}}{m_{1}}=\frac{z-z_{1}}{n_{1}} \text { and } \frac{x-x_{2}}{\ell_{2}}=\frac{y-y_{2}}{m_{2}}=\frac{z-z_{2}}{n_{2}}\)
coplanar then \(\left|\begin{array}{ccc}x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1} \\\ell_{1} & m_{1} & n_{1} \\\ell_{2} & m_{2} & n_{2}\end{array}\right|\) = 0
and \(\left|\begin{array}{ccc}x-x_{1} & y-y_{1} & z-z_{1} \\\ell_{1} & m_{1} & n_{1} \\\ell_{2} & m_{2} & n_{2}\end{array}\right|\) = 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 = \(\left|\begin{array}{llll}a_{1} & b_{1} & c_{1} & d_{1} \\a_{2} & b_{2} & c_{2} & d_{2} \\
a_{3} & b_{3} & c_{3} & d_{3} \\a_{4} & b_{4} & c_{4} & d_{4}\end{array}\right|\) = 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
\(\frac{x-x_{1}}{\ell_{1}}=\frac{y-y_{1}}{m_{1}}=\frac{z-z_{1}}{n_{1}} \text { and } \frac{x-x_{2}}{\ell_{2}}=\frac{y-y_{2}}{m_{2}}=\frac{z-z_{2}}{n_{2}}\)
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
\(\left|\begin{array}{ccc}x-x_{1} & y-y_{1} & z-z_{1} \\\ell_{1} & m_{1} & n_{1} \\
\ell & m & n\end{array}\right|=0 \text { and }\left|\begin{array}{cccc}x-x_{2} & y-y_{2} & z-z_{2} \\\ell_{2} & m_{2} & n_{2} \\\ell & m & n\end{array}\right|=0\)

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 = \(=\frac{1}{6}\left|\begin{array}{llll}x_{1} & y_{1} & z_{1} & 1 \\x_{2} & y_{2} & z_{2} & 1 \\x_{3} & y_{3} & z_{3} & 1 \\x_{4} & y_{4} & z_{4} & 1\end{array}\right|\) = 0

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