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3d formulas

Master 3d through 36 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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3d, every formula

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Distance between two points
$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$
Distance from origin
$d = x_{1}^{2} + y_{1}^{2} + z_{1}^{2}$
Section formula (internal)
$(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n}, \frac{m z _{2} + n z _{1}}{m + n})$
Q1MCQSection internal
The point dividing $(1,0,0)$ and $(0,0,3)$ internally in ratio $1:2$ is:
- A
  $\left(\dfrac23,0,1\right)$
- B
  $\left(\dfrac13,0,2\right)$
- C
  $(1,0,1)$
- D
  $\left(\dfrac12,0,\dfrac32\right)$
Section formula (external)
$(\frac{m x _{2} - n x _{1}}{m - n}, \frac{m y _{2} - n y _{1}}{m - n}, \frac{m z _{2} - n z _{1}}{m - n})$
Q1MCQSection external
The external-division formula uses:
- A
  $\dfrac{mx_2-nx_1}{m-n}$
- B
  $\dfrac{mx_2+nx_1}{m+n}$
- C
  $\dfrac{x_1+x_2}{2}$
- D
  $mx_2-nx_1$
Midpoint
$(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}, \frac{z _{1} + z _{2}}{2})$
Q1MCQMidpoint
The midpoint of $(2,4,6)$ and $(4,0,2)$ is:
- A
  $(3,2,4)$
- B
  $(6,4,8)$
- C
  $(1,2,2)$
- D
  $(2,2,4)$
Centroid of a triangle
$(\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3}, \frac{z _{1} + z _{2} + z _{3}}{3})$
Q1MCQCentroid triangle
The centroid of the triangle with vertices $(1,2,3),(3,2,1),(2,2,2)$ is:
- A
  $(2,2,2)$
- B
  $(6,6,6)$
- C
  $(1,1,1)$
- D
  $(3,2,1)$
Centroid of a tetrahedron
$(\frac{\sum x _{i}}{4}, \frac{\sum y _{i}}{4}, \frac{\sum z _{i}}{4})$
Q1MCQCentroid tetrahedron
The centroid of a tetrahedron divides each vertex-to-opposite-centroid segment in ratio:
- A
  $3:1$
- B
  $2:1$
- C
  $1:1$
- D
  $4:1$
Direction cosines identity
$l^{2} + m^{2} + n^{2} = 1$
d.c.'s from d.r.'s
$l = \frac{a}{a ^{2} + b ^{2} + c ^{2}}, m = \frac{b}{\dots}, n = \frac{c}{\dots}$
Q1MCQd.c. from d.r.
The d.c.'s of the line with d.r.'s $(1,2,2)$ are:
- A
  $\left(\dfrac13,\dfrac23,\dfrac23\right)$
- B
  $(1,2,2)$
- C
  $\left(\dfrac12,1,1\right)$
- D
  $\left(\dfrac19,\dfrac29,\dfrac29\right)$
d.r.'s of a segment
$(x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1})$
Q1MCQd.r. of segment
The d.r.'s of the segment from $(1,2,3)$ to $(4,6,3)$ are:
- A
  $(3,4,0)$
- B
  $(5,8,6)$
- C
  $(3,4,6)$
- D
  $(1,2,3)$
sin²α+sin²β+sin²γ
$sin^{2} α + sin^{2} β + sin^{2} γ = 2$
Q1MCQsin² sum
If a line makes angles $\alpha,\beta,\gamma$ with the axes, then $\sin^{2}\alpha+\sin^{2}\beta+\sin^{2}\gamma$ equals:
- A
  $2$
- B
  $1$
- C
  $3$
- D
  $-1$
cos2α+cos2β+cos2γ
$cos 2 α + cos 2 β + cos 2 γ = - 1$
Q1MCQcos2 sum
$\cos2\alpha+\cos2\beta+\cos2\gamma$ equals:
- A
  $-1$
- B
  $1$
- C
  $2$
- D
  $0$
Angle between two lines (d.c.'s)
$cos θ = ∣ l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} ∣$
Q1MCQAngle (d.c.)
The angle between lines with d.c.'s $(l_1,m_1,n_1),(l_2,m_2,n_2)$ satisfies:
- A
  $\cos\theta=|l_1l_2+m_1m_2+n_1n_2|$
- B
  $\cos\theta=l_1l_2$
- C
  $\sin\theta=l_1l_2+m_1m_2+n_1n_2$
- D
  $\cos\theta=l_1+l_2$
Angle between two lines (d.r.'s)
$cos θ = \frac{∣ a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2} ∣}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}$
Perpendicular lines
$a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$
Q1MCQPerpendicular lines
Lines with d.r.'s $(a_1,b_1,c_1),(a_2,b_2,c_2)$ are perpendicular iff:
- A
  $a_1a_2+b_1b_2+c_1c_2=0$
- B
  $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$
- C
  $a_1+a_2=0$
- D
  $a_1a_2=b_1b_2$
Parallel lines
$\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}$
Q1MCQParallel lines
Lines with d.r.'s $(2,4,6)$ and $(1,2,3)$ are:
- A
  parallel
- B
  perpendicular
- C
  skew
- D
  coincident with the axes
Projection of a segment
$∣ l (x_{2} - x_{1}) + m (y_{2} - y_{1}) + n (z_{2} - z_{1}) ∣$
on a line with d.c.'s (l,m,n)
Q1MCQProjection of segment
The projection of the segment from $(0,0,0)$ to $(3,4,0)$ on the x-axis (d.c.'s $(1,0,0)$) is:
- A
  $3$
- B
  $4$
- C
  $5$
- D
  $0$
Symmetrical form of a line
$\frac{x - x _{1}}{l} = \frac{y - y _{1}}{m} = \frac{z - z _{1}}{n}$
Q1MCQSymmetric form
The line through $(1,2,3)$ with d.r.'s $(2,3,4)$ is:
- A
  $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$
- B
  $\dfrac{x-2}{1}=\dfrac{y-3}{2}=\dfrac{z-4}{3}$
- C
  $2x+3y+4z=0$
- D
  $\dfrac{x+1}{2}=\dfrac{y+2}{3}=\dfrac{z+3}{4}$
Point on a line (parametric)
$(x_{1} + l r, y_{1} + m r, z_{1} + n r)$
Q1MCQPoint on a line
A general point on $\dfrac{x-1}{2}=\dfrac{y}{3}=\dfrac{z+1}{4}=r$ is:
- A
  $(1+2r,\ 3r,\ -1+4r)$
- B
  $(2r,3r,4r)$
- C
  $(1+r,r,-1+r)$
- D
  $(1,0,-1)$ only
Equation of a plane (general)
$a x + b y + cz + d = 0, (a, b, c) are d.r.’s of normal$
Q1MCQPlane general form
$2x+3y-z+5=0$ has normal with d.r.'s:
- A
  $(2,3,-1)$
- B
  $(2,3,5)$
- C
  $(2,3,1)$
- D
  $(5,0,0)$
Plane through a point
$a (x - x_{1}) + b (y - y_{1}) + c (z - z_{1}) = 0$
Q1MCQPlane through a point
The plane through $(1,1,1)$ with normal $(1,2,3)$ is:
- A
  $x+2y+3z=6$
- B
  $x+2y+3z=0$
- C
  $x+y+z=3$
- D
  $x+2y+3z=1$
Plane through three points
$x - x_{1} x_{2} - x_{1} x_{3} - x_{1} y - y_{1} y_{2} - y_{1} y_{3} - y_{1} z - z_{1} z_{2} - z_{1} z_{3} - z_{1} = 0$
Q1MCQPlane through 3 points
The equation of a plane through three non-collinear points is a:
- A
  $3\times3$ determinant $=0$
- B
  $2\times2$ determinant $=0$
- C
  quadratic
- D
  pair of lines
Normal form of a plane
$l x + m y + n z = p$
p = distance from origin
Q1MCQNormal form
In $lx+my+nz=p$, the quantity $p$ represents:
- A
  the perpendicular distance of the plane from the origin
- B
  the x-intercept
- C
  a d.r. of the normal
- D
  the area
Intercept form of a plane
$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$
Q1MCQIntercept form
The plane $\dfrac{x}{2}+\dfrac{y}{3}+\dfrac{z}{4}=1$ has x-intercept:
- A
  $2$
- B
  $3$
- C
  $4$
- D
  $1$
Perpendicular distance (point→plane)
$\frac{∣ a x _{1} + b y _{1} + c z _{1} + d ∣}{a ^{2} + b ^{2} + c ^{2}}$
Distance of plane from origin
$\frac{∣ d ∣}{a ^{2} + b ^{2} + c ^{2}}$
Distance between parallel planes
$\frac{∣ d _{1} - d _{2} ∣}{a ^{2} + b ^{2} + c ^{2}}$
Angle between two planes
$cos θ = \frac{∣ a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2} ∣}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}$
Perpendicular planes
$a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$
Q1MCQPerpendicular planes
Planes $x+2y+2z=1$ and $2x+y-2z=3$ are:
- A
  perpendicular
- B
  parallel
- C
  identical
- D
  at $45^{\circ}$
Angle between line and plane
$sin θ = \frac{∣ a l + bm + c n ∣}{a ^{2} + b ^{2} + c ^{2} l ^{2} + m ^{2} + n ^{2}}$
Q1MCQLine-plane angle
The angle between a line and a plane uses:
- A
  $\sin\theta=\dfrac{|al+bm+cn|}{\sqrt{a^2+b^2+c^2}\sqrt{l^2+m^2+n^2}}$
- B
  $\cos\theta=\dfrac{|al+bm+cn|}{\cdots}$
- C
  $\tan\theta=al+bm+cn$
- D
  $\sin\theta=al+bm+cn$
Line parallel to plane
$a l + bm + c n = 0$
Q1MCQLine parallel to plane
A line with d.r.'s $(l,m,n)$ is parallel to plane $ax+by+cz+d=0$ iff:
- A
  $al+bm+cn=0$
- B
  $\dfrac{a}{l}=\dfrac{b}{m}=\dfrac{c}{n}$
- C
  $al+bm+cn=1$
- D
  $a+b+c=0$
Coplanarity of two lines
$x_{2} - x_{1} a_{1} a_{2} y_{2} - y_{1} b_{1} b_{2} z_{2} - z_{1} c_{1} c_{2} = 0$
Q1MCQCoplanarity of lines
Two lines are coplanar iff the determinant of $(\vec{P_2P_1},\vec{d_1},\vec{d_2})$ equals:
- A
  $0$
- B
  $1$
- C
  $\infty$
- D
  the distance
Shortest distance (skew lines)
$d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}$
Q1MCQShortest distance
The shortest distance between skew lines $\vec r=\vec a_1+\lambda\vec b_1$ and $\vec r=\vec a_2+\mu\vec b_2$ is:
- A
  $\dfrac{|(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)|}{|\vec b_1\times\vec b_2|}$
- B
  $|\vec a_2-\vec a_1|$
- C
  $|\vec b_1\times\vec b_2|$
- D
  $(\vec a_2-\vec a_1)\cdot\vec b_1$
Angle between cube diagonals
$cos θ = \frac{1}{3}$
Q1MCQCube diagonals
The angle between two main diagonals of a cube is:
- A
  $\cos^{-1}\dfrac13$
- B
  $\cos^{-1}\dfrac12$
- C
  $90^{\circ}$
- D
  $60^{\circ}$
Equal-angle d.c.'s
$(\pm \frac{1}{3}, \pm \frac{1}{3}, \pm \frac{1}{3})$
line equally inclined to axes
Q1MCQEqual-angle d.c.'s
A line equally inclined to the three axes has d.c.'s:
- A
  $\left(\pm\dfrac{1}{\sqrt3},\pm\dfrac{1}{\sqrt3},\pm\dfrac{1}{\sqrt3}\right)$
- B
  $(1,1,1)$
- C
  $\left(\dfrac13,\dfrac13,\dfrac13\right)$
- D
  $\left(\dfrac{1}{\sqrt2},\dfrac{1}{\sqrt2},0\right)$
Area of a triangle (vectors)
$Area = \frac{1}{2} A B \times A C$

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3d formulas

3d, every formula

Distance between two points

Distance from origin

Section formula (internal)

Section formula (external)

Midpoint

Centroid of a triangle

Centroid of a tetrahedron

Direction cosines identity

d.c.'s from d.r.'s

d.r.'s of a segment

sin²α+sin²β+sin²γ

cos2α+cos2β+cos2γ

Angle between two lines (d.c.'s)

Angle between two lines (d.r.'s)

Perpendicular lines

Parallel lines

Projection of a segment

Symmetrical form of a line

Point on a line (parametric)

Equation of a plane (general)

Plane through a point

Plane through three points

Normal form of a plane

Intercept form of a plane

Perpendicular distance (point→plane)

Distance of plane from origin

Distance between parallel planes

Angle between two planes

Perpendicular planes

Angle between line and plane

Line parallel to plane

Coplanarity of two lines

Shortest distance (skew lines)

Angle between cube diagonals

Equal-angle d.c.'s

Area of a triangle (vectors)

More JEE Advanced Mathematics formula sheets

Other ways to revise this chapter

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