3d formulas

Every formula JEE Advanced expects you to know for 3d, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 36
Subject: Mathematics
Exam: JEE Advanced
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3d, every formula

36 formulas, typeset and free. Print it, or keep it open beside your practice.

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)
$(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n}, \frac{m z _{2} + n z _{1}}{m + n})$
Section formula (external)
$(\frac{m x _{2} - n x _{1}}{m - n}, \frac{m y _{2} - n y _{1}}{m - n}, \frac{m z _{2} - n z _{1}}{m - n})$
Midpoint
$(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}, \frac{z _{1} + z _{2}}{2})$
Centroid of a triangle
$(\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3}, \frac{z _{1} + z _{2} + z _{3}}{3})$
Centroid of a tetrahedron
$(\frac{\sum x _{i}}{4}, \frac{\sum y _{i}}{4}, \frac{\sum z _{i}}{4})$
Direction cosines identity
$l^{2} + m^{2} + n^{2} = 1$
d.c.'s from d.r.'s
$l = \frac{a}{a ^{2} + b ^{2} + c ^{2}}, m = \frac{b}{\dots}, n = \frac{c}{\dots}$
d.r.'s of a segment
$(x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1})$
sin²α+sin²β+sin²γ
$sin^{2} α + sin^{2} β + sin^{2} γ = 2$
cos2α+cos2β+cos2γ
$cos 2 α + cos 2 β + cos 2 γ = - 1$
Angle between two lines (d.c.'s)
$cos θ = ∣ l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} ∣$
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
$a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$
Parallel lines
$\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}$
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)
Symmetrical form of a line
$\frac{x - x _{1}}{l} = \frac{y - y _{1}}{m} = \frac{z - z _{1}}{n}$
Point on a line (parametric)
$(x_{1} + l r, y_{1} + m r, z_{1} + n r)$
Equation of a plane (general)
$a x + b y + cz + d = 0, (a, b, c) are d.r.’s of normal$
Plane through a point
$a (x - x_{1}) + b (y - y_{1}) + c (z - z_{1}) = 0$
Plane through three points
$x - x_{1} x_{2} - x_{1} x_{3} - x_{1} y - y_{1} y_{2} - y_{1} y_{3} - y_{1} z - z_{1} z_{2} - z_{1} z_{3} - z_{1} = 0$
Normal form of a plane
$l x + m y + n z = p$
p = distance from origin
Intercept form of a plane
$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$
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
$a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$
Angle between line and plane
$sin θ = \frac{∣ a l + bm + c n ∣}{a ^{2} + b ^{2} + c ^{2} l ^{2} + m ^{2} + n ^{2}}$
Line parallel to plane
$a l + bm + c n = 0$
Coplanarity of two lines
$x_{2} - x_{1} a_{1} a_{2} y_{2} - y_{1} b_{1} b_{2} z_{2} - z_{1} c_{1} c_{2} = 0$
Shortest distance (skew lines)
$d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}$
Angle between cube diagonals
$cos θ = \frac{1}{3}$
Equal-angle d.c.'s
$(\pm \frac{1}{3}, \pm \frac{1}{3}, \pm \frac{1}{3})$
line equally inclined to axes
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)

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