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Class 11 Maths Notes

Three Dimensional Geometry Class 11 Notes

Complete, exam-ready notes on introduction to three-dimensional geometry: how the three coordinate axes divide space into octants, coordinates of a point in 3D, the distance and section formulas, direction cosines and direction ratios of a line, and the angle between two lines using direction ratios — written for CBSE boards and JEE revision.

Class11SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is this chapter about in one line?

Three-dimensional geometry extends the coordinate plane into space: you locate points with three coordinates and measure distances, ratios and angles using direction cosines.

Coordinate Axes and Planes in Space

Coordinate axes

Three mutually perpendicular lines — the x-axis, y-axis and z-axis — meet at the origin O(0, 0, 0). Any point P in space is located by the ordered triple (x, y, z).

  • The xy-plane, yz-plane and zx-plane divide space into eight octants.
  • In the first octant, x > 0, y > 0 and z > 0.
  • The octant sign pattern (±, ±, ±) identifies all eight octants — a frequent MCQ topic.
  • The z-coordinate extends the 2D plane vertically into 3D space.

Distance Between Two Points

3D distance formula

The distance between P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is: PQ=(x2−x1)2+(y2−y1)2+(z2−z1)2PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}. This is the natural extension of the 2D distance formula with the third dimension added.

Distance from origin

For any point P(x, y, z), the distance from the origin O(0, 0, 0) is: OP=x2+y2+z2OP = \sqrt{x^2 + y^2 + z^2}.

Section Formula

Internal division

If point R divides the segment joining P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally in the ratio m : n, the coordinates of R are: (mx2+nx1m+n,  my2+ny1m+n,  mz2+nz1m+n)\left(\frac{mx_2 + nx_1}{m + n},\; \frac{my_2 + ny_1}{m + n},\; \frac{mz_2 + nz_1}{m + n}\right).

Midpoint

The midpoint (m = n) of segment PQ is: (x1+x22,  y1+y22,  z1+z22)\left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2},\; \frac{z_1 + z_2}{2}\right). This is a special case of the section formula with m = n = 1.

  • External division: replace m + n with m − n in the denominator, and flip the sign of n.
  • The section formula in 3D mirrors the 2D version — just add the z-component.

Direction Cosines and Direction Ratios

Direction cosines

The cosines of the angles α, β, γ that a line makes with the positive x-, y- and z-axes are called direction cosines, denoted l, m, n. They always satisfy: l2+m2+n2=1l^2 + m^2 + n^2 = 1.

Direction ratios

a,b,ca, b, c

Any three numbers a, b, c proportional to the direction cosines are direction ratios of the line: l=aa2+b2+c2,m=ba2+b2+c2,n=ca2+b2+c2l = \frac{a}{\sqrt{a^2 + b^2 + c^2}},\quad m = \frac{b}{\sqrt{a^2 + b^2 + c^2}},\quad n = \frac{c}{\sqrt{a^2 + b^2 + c^2}}. Direction ratios are not unique — any scalar multiple works.

Practical use

For a line through P(x₁, y₁, z₁) and Q(x₂, y₂, z₂), the direction ratios are simply (x₂ − x₁, y₂ − y₁, z₂ − z₁). Normalise to get direction cosines.

Angle Between Two Lines

Angle using direction cosines

If two lines have direction cosines (l₁, m₁, n₁) and (l₂, m₂, n₂), the angle θ between them satisfies: cos⁡θ=l1l2+m1m2+n1n2\cos\theta = l_1 l_2 + m_1 m_2 + n_1 n_2. For the acute angle, take the absolute value.

Angle using direction ratios

For direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂): cos⁡θ=∣a1a2+b1b2+c1c2∣a12+b12+c12  a22+b22+c22\cos\theta = \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}}.

  • Lines are parallel when their direction ratios are proportional: a₁/a₂ = b₁/b₂ = c₁/c₂.
  • Lines are perpendicular when a₁a₂ + b₁b₂ + c₁c₂ = 0.

Collinearity of Three Points

Collinearity test

Three points A(x₁, y₁, z₁), B(x₂, y₂, z₂) and C(x₃, y₃, z₃) are collinear if and only if their direction ratios are proportional — equivalently, x2−x1x3−x1=y2−y1y3−y1=z2−z1z3−z1\frac{x_2 - x_1}{x_3 - x_1} = \frac{y_2 - y_1}{y_3 - y_1} = \frac{z_2 - z_1}{z_3 - z_1}. If all three ratios are equal, the points lie on a single line.

  • An equivalent check: AB + BC = AC (sum of distances).
  • If the ratios are not equal, the three points form a triangle with non-zero area in 3D.
  • Collinearity is a prerequisite for applying the section formula — ensure the points are collinear before dividing a segment.

Solved Examples

Example: Find the distance between the points P(1, −3, 4) and Q(3, −5, 2).

Solution: PQ = √[(3 − 1)² + (−5 + 3)² + (2 − 4)²] = √[4 + 4 + 4] = √12 = 2√3 units.

Example: Find the direction cosines of the line joining A(1, 2, −3) and B(3, −2, 1).

Solution: Direction ratios = (3−1, −2−2, 1−(−3)) = (2, −4, 4). The magnitude is √(4 + 16 + 16) = √36 = 6. So l = 2/6 = 1/3, m = −4/6 = −2/3, n = 4/6 = 2/3. Check: l² + m² + n² = 1/9 + 4/9 + 4/9 = 1.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Distance between two points

(x2−x1)2+(y2−y1)2+(z2−z1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

Internal section formula

(mx2+nx1m+n,  my2+ny1m+n,  mz2+nz1m+n)\left(\frac{mx_2 + nx_1}{m + n},\; \frac{my_2 + ny_1}{m + n},\; \frac{mz_2 + nz_1}{m + n}\right)

Midpoint formula

(x1+x22,  y1+y22,  z1+z22)\left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2},\; \frac{z_1 + z_2}{2}\right)

Direction cosine identity

l2+m2+n2=1l^2 + m^2 + n^2 = 1

Direction ratios to cosines

l=aa2+b2+c2l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}

Angle between two lines

cos⁡θ=l1l2+m1m2+n1n2\cos\theta = l_1 l_2 + m_1 m_2 + n_1 n_2

Distance from origin

x2+y2+z2\sqrt{x^2 + y^2 + z^2}

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

  • Direction cosines satisfy l² + m² + n² = 1 always — use this to verify your answer.
  • Direction ratios are not unique; only their ratios matter.
  • The 3D section formula mirrors 2D — just add the z-component.
  • Parallel lines: direction ratios are proportional. Perpendicular lines: a₁a₂ + b₁b₂ + c₁c₂ = 0.
  • Octant identification from sign of coordinates is a common MCQ topic.
  • Distance from origin is a special case: √(x² + y² + z²).
  • For angle between lines, normalise direction ratios to cosines first or use the full formula.
  • The midpoint formula is the section formula with m = n = 1 — know both.

FAQ

Common questions

What are direction cosines?

Direction cosines are the cosines of the angles a line makes with the positive x-, y- and z-axes, denoted l, m, n. They always satisfy l² + m² + n² = 1.

How do you find the distance between two points in 3D?

Use the extended distance formula: PQ = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²], which adds the z-component to the 2D version.

What is the section formula in 3D?

A point dividing the segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) in ratio m : n has coordinates ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)).

How do direction ratios differ from direction cosines?

Direction ratios are any three numbers proportional to the direction cosines — they are not unique. Direction cosines are the normalised values that satisfy l² + m² + n² = 1.

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Class 11 Maths NotesAll Class 11 & 12 NotesClass 12 Maths NotesMaths Tutor in MohaliStraight Lines NotesConic Sections NotesLimits and Derivatives Notes

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