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Important Three Dimensional Geometry Formulas for JEE Maths

Three-dimensional geometry plays a major role as a lot of questions are included in the JEE exam. Here is a list of all the three-dimensional geometry formulas which will help students to go through and revise them quickly before the exam.

Three-dimensional Geometry Formulas

1. Vector representation of a point: Position vector of a point P(x, y, z) is xi^+yj^+zk^x\hat{i}+y\hat{j}+z\hat{k}

2.Distanceformula:(x1−x2)2+(y1−y2)2+(z1−z2)2\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}} , AB = ∣OB⃗−OA⃗∣\left | \vec{OB} -\vec{OA}\right |

3. Distance of P from coordinate axes: PA = (y2+z2)\sqrt{(y^{2}+z^{2})}

PB = (z2+x2)\sqrt{(z^{2}+x^{2})}

PC = (x2+y2)\sqrt{(x^{2}+y^{2})}

4.Section Formula:

x = mx2+nx1m+n\frac{mx_{2}+nx_{1}}{m+n}

y = my2+ny1m+n\frac{my_{2}+ny_{1}}{m+n}

z = mz2+nz1m+n\frac{mz_{2}+nz_{1}}{m+n}

Midpoint:

x = x1+x22\frac{x_{1}+x_{2}}{2}

y = y1+y22\frac{y_{1}+y_{2}}{2}

z = z1+z22\frac{z_{1}+z_{2}}{2}

5. Centroid of a triangle:

G = (x1+x2+x33,y1+y2+y33,z1+z2+z33)\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 )

6. Incentre of triangle ABC: (ax1+bx2+cx3a+b+c,ay1+by2+cy3a+b+c,az1+bz2+cz3a+b+c)\left ( \frac{ax_{1}+bx_{2}+cx_{3}}{a+b+c},\frac{ay_{1}+by_{2}+cy_{3}}{a+b+c},\frac{az_{1}+bz_{2}+cz_{3}}{a+b+c} \right )

7. Centroid of a tetrahedron: (x1+x2+x3+x44,y1+y2+y3+y44,z1+z2+z3+z44)\left ( \frac{x_{1}+x_{2}+x_{3}+x_{4}}{4},\frac{y_{1}+y_{2}+y_{3}+y_{4}}{4},\frac{z_{1}+z_{2}+z_{3}+z_{4}}{4} \right )

8. Direction cosines and direction ratios:

(i) Direction cosines: let α, β, γ be the angles which a directed line makes with the positive directions of the axes of x, y and z respectively, then cos α, cos β and cos γ are called the direction cosines of the line. The direction cosines are usually denoted by (l, m, n).

Therefore l = cos α, m = cos β, n = cos γ.

(ii) l2+m2+n2 = 1

(iii) If a, b, c are the direction ratios of any line L then ai^+bj^+ck^a\hat{i}+b\hat{j}+c\hat{k} will be a vector parallel to the line L.

(iv) If l, m, n are the direction cosines of any line L then li^+mj^+nk^l\hat{i}+m\hat{j}+n\hat{k} is a unit vector parallel to the line L.

(v) If l, m, n be the direction cosines and a, b, c be the direction ratios of a vector, then

l = aa2+b2+c2\frac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}

m = ba2+b2+c2\frac{b}{\sqrt{a^{2}+b^{2}+c^{2}}}

n = ca2+b2+c2\frac{c}{\sqrt{a^{2}+b^{2}+c^{2}}}

or l = −aa2+b2+c2\frac{-a}{\sqrt{a^{2}+b^{2}+c^{2}}},

m = −ba2+b2+c2\frac{-b}{\sqrt{a^{2}+b^{2}+c^{2}}}

n = −ca2+b2+c2\frac{-c}{\sqrt{a^{2}+b^{2}+c^{2}}}

(vi) If OP = r, the direction cosines of OP are l, m, n then the coordinates of P are ( lr, mr, nr).

If the direction cosines of the line AB are l, m, n, |AB| = r and the coordinates of A is (x1, y1, z1) then the coordinates of B are given as (x1+rl, y1+rm, z1+rn).

(vii) If the coordinates P and Q are (x1, y1, z1) and (x2, y2, z2) then the direction ratios of line PQ are a = x2-x1 , b = y2-y1 and c = z2-z1 and the direction cosines of line PQ are l = x2−x1∣PQ∣\frac{x_{2}-x_{1}}{\left | PQ \right |}, m = y2−y1∣PQ∣\frac{y_{2}-y_{1}}{\left | PQ \right |} and n = z2−z1∣PQ∣\frac{z_{2}-z_{1}}{\left | PQ \right |}

(vii) Direction cosines of the x-axis is (1, 0, 0).

Direction cosines of the y-axis is (0, 1, 0).

Direction cosines of the z-axis is (0, 0, 1).

9. Angle between two line segments:

cos θ = ∣a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22∣\left | \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}}} \right |

The line will be perpendicular if a1a2+b1b2+c1c2 = 0, and parallel if a1/a2 = b1/b2 = c1/c2.

10. Projection of a line segment on a line:

If P(x1, y1, z1) and Q(x2, y2, z2) then the projection of PQ on a line having direction cosines l, m, n is |l(x2-x1)+m(y2-y1)+n(z2-z1)|

11. Equation of a plane: General form: ax+by+cz+d = 0, where a, b, c are not all zero, a, b, c, d ∈ R.

(i) Normal form: lx+my+nz = p

(ii) Plane through the point (x1, y1, z1): a(x-x1)+b(y-y1)+c(z-z1) = 0

(iii) Intercept form: xa+yb+zc=1\frac{x}{a}+\frac{y}{b}+\frac{z}{c} = 1

(iv) vector form: (r⃗−a⃗).n⃗=0(\vec{r}-\vec{a} ).\vec{n}= 0 or r⃗. n⃗=a⃗. n⃗\vec{r}.\: \vec{n}= \vec{a}.\: \vec{n}

(v) Planes parallel to the axes :

(a) plane parallel to X-axis is by+cz+d = 0

(b) plane parallel to Y-axis is ax+cz+d = 0

(c) plane parallel to Z-axis is ax+by+d = 0

(vi) Plane through origin: Equation of the plane passing through the origin is ax+by+cz = 0.

(vii) Transformation of the equation of a plane to the normal form: ax+by+cz-d = 0in normal form is ax±a2+b2+c2+by±a2+b2+c2+cz±a2+b2+c2=d±a2+b2+c2\frac{ax}{\pm \sqrt{a^{2}+b^{2}+c^{2}}}+\frac{by}{\pm \sqrt{a^{2}+b^{2}+c^{2}}}+\frac{cz}{\pm \sqrt{a^{2}+b^{2}+c^{2}}} = \frac{d}{\pm \sqrt{a^{2}+b^{2}+c^{2}}}

(viii) Any plane parallel to the given plane ax+by+cz+d = 0 is ax+by+cz+λ = 0.

Distance between ax+by+cz+d1 = 0 and ax+by+cz+d2 = 0 is ∣d1−d2∣a2+b2+c2\frac{\left | d_{1}-d_{2} \right |}{\sqrt{a^{2}+b^{2}+c^{2}}}

(ix) A plane ax+by+cz+d = 0 divides the line segment joining (x1, y1, z1) and (x2, y2, z2) in the ratio (−ax1+by1+cz1+dax2+by2+cz2+d)\left ( -\frac{ax_{1}+by_{1}+cz_{1}+d}{ax_{2}+by_{2}+cz_{2}+d}\right )

(x) Coplanarity of four points: The points A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) and D(x4, y4, z4) are coplanar if ∣x2−x1y2−y1z2−z1x3−x1y3−y1z3−z1x4−x1y4−y1z4−z1∣=0\begin{vmatrix} x_{2}-x_{1}& y_{2}-y_{1} & z_{2}-z_{1}\\ x_{3}-x_{1} & y_{3}-y_{1} & z_{3}-z_{1} \\ x_{4}-x_{1}& y_{4}-y_{1} & z_{4}-z_{1} \end{vmatrix}= 0

12. A point and a plane:

(i) distance of the point (x’, y’, z’) from the plane ax+by+cz+d = 0 is given by ax‘+by‘+cz‘+da2+b2+c2\frac{ax^{‘}+by^{‘}+cz^{‘}+d}{\sqrt{a^{2}+b^{2}+c^{2}}}

(ii) Length of the perpendicular from a point a⃗\vec{a} to the plane r⃗.n⃗=d\vec{r}.\vec{n} = d is given by p = ∣a⃗. n⃗−d∣∣n⃗∣\frac{\left |\vec{a} .\: \vec{n} -d\right |}{\left | \vec{n} \right |}.

(iii) Foot (x’, y’, z’) of perpendicular drawn from the point (x1, y1, z1) to the plane ax+by+cz+d = 0 is given by x‘−x1a=y‘−y1b=z‘−z1c=−ax1+by1+cz1+da2+b2+c2\frac{x^{‘}-x_{1}}{a} = \frac{y^{‘}-y_{1}}{b} =\frac{z^{‘}-z_{1}}{c}= -\frac{ax_{1}+by_{1}+cz_{1}+d}{a^{2}+b^{2}+c^{2}}

(iv) To find image of a point with respect to a plane:

Let P (x1, y1, z1) be a given point and ax+by+cz+d = 0 is given plane. Let (x’, y’, z’) is the image point. Then x‘−x1a=y‘−y1b=z‘−z1c=−2(ax1+by1+cz1+d)a2+b2+c2\frac{x^{‘}-x_{1}}{a} = \frac{y^{‘}-y_{1}}{b} =\frac{z^{‘}-z_{1}}{c}= -2\frac{(ax_{1}+by_{1}+cz_{1}+d)}{a^{2}+b^{2}+c^{2}}.

(v) The distance between two parallel planes ax+by+cz+d = 0 and ax+by+cz+d’ = 0 is given by ∣d−d’∣a2+b2+c2\frac{\left | d-d{}’ \right |}{\sqrt{a^{2}+b^{2}+c^{2}}}

13. Angle between two planes:

(i) cos⁡θ=∣aa’+bb’+cc’a2+b2+c2a’2+b’2+c’2∣\cos \theta =\left | \frac{aa{}’+bb{}’+cc{}’}{\sqrt{a^{2}+b^{2}+c^{2}}\sqrt{a{}’^{2}+b{}’^{2}+c{}’^{2}}}\right |

(ii) Planes are perpendicular if aa’+bb’+cc’ = 0 and parallel if a/a’ = b/b’ = c/c’.

(iii) The angle θ between the planes r⃗.n1⃗=d1\vec{r}.\vec{n_{1}} = d_{1} and r⃗.n2⃗=d2\vec{r}.\vec{n_{2}} = d_{2} is given by cos θ = n1⃗.n2⃗∣n1⃗∣.∣n2⃗∣\frac{\vec{n_{1}}.\vec{n_{2}} }{\left | \vec{n_{1}} \right |.\left | \vec{n_{2}} \right |}

(iv) Planes are perpendicular if n1⃗.n2⃗\vec{n_{1}}.\vec{n_{2}} = 0 and planes are parallel if n1⃗=λn2⃗\vec{n_{1}}=\lambda \vec{n_{2}}, λ is a scalar.

14. Angle bisectors:

(i) The equations of a planes bisecting the angle between two given planes a1x+b1y+c1z+d1 = 0 and a2x+b2y+c2z+d2 = 0 are a1x+b1y+c1z+d1a12+b12+c12\frac{a_{1}x+b_{1}y+c_{1}z+d_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}} = ±a2x+b2y+c2z+d2a22+b22+c22\pm \frac{a_{2}x+b_{2}y+c_{2}z+d_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}

(ii) Bisector of acute or obtuse angle: First make both the constant terms positive. Then a1a2+b1b2+c1c2 > 0 ⇒ origin lies on obtuse angle.

a1a2+b1b2+c1c2 < 0 ⇒ origin lies on acute angle.

15. Area of a triangle:

Let A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) be the vertices of a triangle, then Δ = Δx2+Δy2+Δz2\sqrt{\Delta_{x}^{2}+\Delta _{y}^{2}+\Delta _{z}^{2}} where Δx=12∣y1z11y2z21y3z31∣\Delta _{x} = \frac{1}{2}\begin{vmatrix} y_{1} & z_{1} & 1\\ y_{2} &z_{2} & 1\\ y_{3}& z_{3} & 1 \end{vmatrix}, Δy=12∣z1x11z2x21z3x31∣\Delta _{y} = \frac{1}{2}\begin{vmatrix} z_{1} & x_{1} & 1\\ z_{2} &x_{2} & 1\\ z_{3}& x_{3} & 1 \end{vmatrix} and Δz=12∣x1y11x2y21x3y31∣\Delta _{z} = \frac{1}{2}\begin{vmatrix} x_{1} & y_{1} & 1\\ x_{2} &y_{2} & 1\\ x_{3}& y_{3} & 1 \end{vmatrix}

Vector method: from two vector AB⃗\vec{AB} and AC⃗\vec{AC} the area is given by 12∣AB⃗×AC⃗∣\frac{1}{2}\left |\vec{AB} \times \vec{AC} \right |

16. Volume of a tetrahedron:

Volume of a tetrahedron with vertices A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) and D(x4, y4, z4) is given by V = 16∣x1y1z11x2y2z21x3y3z31x4y4z41∣\frac{1}{6}\begin{vmatrix} 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{vmatrix}

17. Equation of a line:

(i) A straight line is the intersection of two planes. It is represented by two planes a1x+b1y+c1z+d1 = 0 and a2x+b2y+c2z+d2 = 0 .

(ii) Symmetric form: (x−x1)a=(y−y1)b=(z−z1)c=r\frac{(x-x_{1})}{a} = \frac{(y-y_{1})}{b}= \frac{(z-z_{1})}{c} = r

(iii) vector equation:r⃗=a⃗+λb⃗\vec{r} = \vec{a}+\lambda \vec{b}

(iv) Reduction of cartesian form of equation of a line to vector form and vice versa x−x1a=y−y1b=z−z1c\frac{x-x_{1}}{a} = \frac{y-y_{1}}{b}= \frac{z-z_{1}}{c} ⇔r⃗=(x1i^+y1j^+z1k^)+λ(ai^+bj^+ck^)\vec{r} = (x_{1}\hat{i}+y_{1}\hat{j}+z_{1}\hat{k})+\lambda (a\hat{i}+b\hat{j}+c\hat{k})

18. To find image of a point with respect to a line:

Let L = x−x2a=y−y2b=z−z2c\frac{x-x_{2}}{a} = \frac{y-y_{2}}{b}= \frac{z-z_{2}}{c} is a given line.

Let (x’, y’, z’) is the image of the point (x1, y1, z1) with respect to the line L. Then

(i) a(x1-x’)+b(y1-y’)+C(z1-z’) = 0

(ii) x1+x’2−x2a=y1+y’2−y2b=z1+z’2−z2c=λ\frac{\frac{x_{1}+x{}’}{2}-x_{2}}{a}= \frac{\frac{y_{1}+y{}’}{2}-y_{2}}{b}= \frac{\frac{z_{1}+z{}’}{2}-z_{2}}{c} = \lambda

From (ii) get the value of x’, y’, z’ in terms of λ as x’= 2aλ+2x2-x1, y’= 2bλ-2y2-y1,

z’= 2cλ+2z2-z1 . Then put the values of x’, y’, z’ in 9i) get λ and substitute the value of λ to get (x’, y’,z’).

19. Angle between a line and a plane:

(i) If θ is the angle between a line x−x1l=y−y1m=z−z1n\frac{x-x_{1}}{l}= \frac{y-y_{1}}{m}=\frac{z-z_{1}}{n} and the plane ax+by+cz+d = 0, then sin⁡θ=∣al+bm+cna2+b2+c2l2+m2+n2∣\sin \theta = \left | \frac{al+bm+cn}{\sqrt{a^{2}+b^{2}+c^{2}}\sqrt{l^{2}+m^{2}+n^{2}}} \right |

(ii) Vector form : If θ is the angle between a line r⃗=a⃗+λb⃗\vec{r}= \vec{a}+\lambda \vec{b} and r⃗.n⃗=d\vec{r}.\vec{n }= d then sin θ = b⃗.n⃗∣b⃗∣∣n⃗∣\frac{\vec{b}.\vec{n}}{\left | \vec{b} \right |\left | \vec{n} \right |}

(iii) condition for perpendicularity l/a = m/b = n/c , b⃗×n⃗=0\vec{b}\times \vec{n}= 0

(iv) condition for parallel : al+bm+cn = o, b⃗.n⃗=0\vec{b}.\vec{n}= 0

20. Condition for a line to lie in a plane:

(i) cartesian form: Line x−x1l=y−y1m=z−z1n\frac{x-x_{1}}{l}= \frac{y-y_{1}}{m}=\frac{z-z_{1}}{n} would lie in a plane ax+by+cz+d = 0, if ax1+by1+cz1+d = 0 and al+bm+cn = 0

(ii) Vector form: Line r⃗=a⃗+λb\vec{r}= \vec{a}+\lambda b would lie in the plane r⃗.n⃗=d\vec{r}.\vec{n}= d if b⃗.n⃗=0\vec{b}.\vec{n}= 0 and a⃗.n⃗=d\vec{a}.\vec{n}= d

21. Skew lines:

(i) The straight lines which are not parallel and non-coplanar are called skew lines.

If Δ=∣α‘−αβ’−βγ’−γlmnl’m’n’∣≠0\Delta = \begin{vmatrix} \alpha ^{‘} -\alpha & \beta {}’-\beta &\gamma {}’-\gamma \\ l & m &n \\ l’& m’&n’ \end{vmatrix} \neq 0 , then the lines are skew.

(ii) Shortest distance:

SD = ∣α‘−αβ’−βγ’−γlmnl’m’n’∣∑(mn’−m’n)2\frac{\begin{vmatrix} \alpha ^{‘} -\alpha & \beta {}’-\beta &\gamma {}’-\gamma \\ l & m &n \\ l’& m’&n’ \end{vmatrix} }{\sqrt{\sum (mn’-m’n)^{2}}}

(iii) Vector form: For lines r⃗=a1⃗+λb1⃗\vec{r} = \vec{a_{1}}+\lambda \vec{b_{1}} and r⃗=a2⃗+λb2⃗\vec{r} = \vec{a_{2}}+\lambda \vec{b_{2}} to be skew (b1⃗×b2⃗).(a2⃗−a1⃗)≠0(\vec{b_{1}}\times \vec{b_{2}}).(\vec{a_{2}}-\vec{a_{1}})\neq 0

(iv) Shortest distance between lines r⃗=a⃗+λb⃗\vec{r} = \vec{a}+\lambda \vec{b}

and r⃗=a2⃗+μb⃗\vec{r} = \vec{a_{2}}+\mu \vec{b} is d=∣(a2⃗−a⃗1)×b⃗∣b⃗∣∣d= \left | \frac{(\vec{a_{2}}-\vec{a}_{1})\times \vec{b}}{\left | \vec{b} \right |} \right |

22. Family of planes:

(i) Any plane through the intersection of a1x+b1y+c1z+d1 = 0 and a2x+b2y+c2z+d2 = 0 is a1x+b1y+c1z+d1+λ(a2x+b2y+c2z+d2) = 0

(ii) The equation of plane passing through the intersection of the planes r⃗.n1⃗=d1\vec{r}.\vec{n_{1}}= d_{1} and r⃗.n2⃗=d2\vec{r}.\vec{n_{2}}= d_{2} is r⃗.(n1⃗+λn2⃗)=d1+λd2\vec{r}.(\vec{n_{1}}+\lambda \vec{n_{2}})=d_{1}+ \lambda d_{2} where λ is an arbitrary scalar.

23. Sphere:

General equation of a sphere is x2+y2+z2+2ux+2vy+2wz+d = 0. (-u, -v, -w) is the centre and u2+v2+w2−d\sqrt{u^{2}+v^{2}+w^{2}-d} is the radius of the sphere.

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