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Three Dimensional Geometry revision notes

Every idea in the Three Dimensional Geometry chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 22 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Coordinates, distance and section formula

Distance and coordinate planes
7× asked, last 2026
PQ=(x2−x1)2+(y2−y1)2+(z2−z1)2,dx-axis=y2+z2PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2},\qquad d_{x\text{-axis}}=\sqrt{y^2+z^2}
xy-plane: z=0z=0; yz-plane: x=0x=0; zx-plane: y=0y=0.
xxyyzzP(x,y,z)P(x,y,z)y2+z2\sqrt{y^2+z^2}
Distance of PP from the xx-axis is to the foot (x,0,0)(x,0,0): y2+z2\sqrt{y^2+z^2}, never more than OPOP
  • A line meets a coordinate plane where that coordinate is 0: for the yz-plane put x=0x=0 in the general point. The length PQPQ along the line is ∣t∣ ∣b⃗∣|t|\,|\vec b|, not ∣xP∣|x_P|.
  • Not ∣x∣|x| or r2+x2\sqrt{r^2+x^2} for the distance from the xx-axis. That distance ignores xx: y2+z2=r2−x2\sqrt{y^2+z^2}=\sqrt{r^2-x^2}, so it can never exceed OP=rOP=r.
Section formula and centroid
5× asked, last 2026
(mx2±nx1m±n, my2±ny1m±n, mz2±nz1m±n),G=A+B+C3\left(\frac{mx_2\pm nx_1}{m\pm n},\ \frac{my_2\pm ny_1}{m\pm n},\ \frac{mz_2\pm nz_1}{m\pm n}\right),\qquad G=\frac{A+B+C}{3}
Divides ABAB in m:nm:n; plus sign internal, minus external. Midpoint: m=nm=n. The xy-plane cuts ABAB in ratio −z1:z2-z_1:z_2.
AA (z1=−1z_1=-1)BB (z2=2z_2=2)PPxyxy-plane
The xyxy-plane (z=0z=0) cuts ABAB at PP with AP:PB=−z1:z2=1:2AP:PB=-z_1:z_2=1:2 (below the plane, then above)
  • Quick method. Centroid from a vertex RR and the midpoint MM of the opposite side: G=(2M+R)/3G=(2M+R)/3, no need for the other two vertices. R(−1,4,2)R(-1,4,2), M(2,1,2)M(2,1,2) give G(1,2,2)G(1,2,2).

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Practise Three Dimensional Geometry

20 more concepts in 5 topics

Direction cosines and direction ratios; Angle between lines; Equation of a line; Foot, image and distance from a line; and Skew lines and shortest distance. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Direction cosines and direction ratios

  • Direction cosines4×
  • Direction ratios3×
  • Projection of a segment on a line3×
  • DCs from two equations4×

Angle between lines

  • Angle between two lines8×
  • Line perpendicular to two lines11×

Equation of a line

  • Line through a point along a direction20×
  • Line through two points7×
  • Point at a given distance on a line9×
  • Intersection of two lines18×
  • Line meeting two given lines9×

Foot, image and distance from a line

  • Foot of the perpendicular on a line19×
  • Image of a point in a line22×
  • Distance of a point from a line15×
  • Distance along a given direction8×
  • Triangle and parallelogram areas15×

Skew lines and shortest distance

  • Parallel, intersecting or skew3×
  • Shortest distance between skew lines41×
  • Distance between parallel lines2×
  • Line of shortest distance4×

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About 295 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

Practise Three Dimensional Geometry

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