CBSE Class 12 Mathematics Notes Chapter 11 Three-Dimensional Geometry

Three-dimensional geometry (3D geometry) is the mathematics of shapes in three-dimensional space, which consists of three coordinates, namely x-coordinate, y-coordinate, and z-coordinate.

• It is the representation of a line or a plane in 3D space.
• In three-dimensional space, the three coordinates are required to find the exact location of a point.
• A cartesian coordinate system consists of three axes, the x-axis, the y-axis, and the z-axis.
• These axes are mutually perpendicular to each other and its point of intersection is the origin O.
• The axes of three-dimensional geometry divide the space into eight octants.
• Abscissa represents the distance of a point along the x-axis from the origin.
• Ordinate represents the y value, which is the perpendicular distance of the point from the x-axis and is parallel to the y-axis.

CBSE Class 12 Mathematics Notes for Chapter 11 Three-Dimensional Geometry are given in the article below for easy preparation and understanding of the concepts involved.

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Direction Cosines

• Let α, ꞵ, and γ be the angles that a directed line segment OP makes with the positive directions of the coordinate axes OX, OY, and OZ respectively.
• Then cos α, cos ꞵ, and cos γ are known as the direction cosines of OP.
• Generally, they are denoted by letters l, m, and n respectively.

Where

• l = cos α
• m = cos ꞵ
• n = cos γ

Direction Cosines

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Properties of Direction Cosines

• If OP is a directed line segment with direction cosines l, m, and n such that OP = r, then the coordinates of P are (lr, mr, nr).
• The sum of squares of the direction cosines is always unity i.e. 

l² + m² + n² = 1

• Parallel lines have the same direction cosines.
• The direction cosines of a line are always unique.
• 0 ≤ α, ꞵ, γ ≤ π.

Direction Ratios

• Let l, m, and n be the direction cosines of a line and a, b, and c be three numbers such that l/a = m/b, and n/c.
• Then, the direction ratios of the line are proportional to a, b, and c.

Relation Between Direction Cosines and Direction Ratios

• If the direction ratios of a line are proportional to a, b, and c then its direction cosines are
• l = ± a/√(a² + b² + c²)
• m = ± b/√(a² + b² + c²)
• n = ± c/√(a² + b² + c²)

Angle Between Two Lines

• If two lines whose direction cosines are (l₁, m₁, and n₁) and (l₂, m₂, n₂), then angle θ between them is given by

cos θ = |l₁l₂ + m₁m₂ + n₁n₂|

• If the direction ratio is given by (a₁, b₁, c₁) and (a₂, b₂, c₂) respectively, then

cos θ = |(a₁a₂ + b₁b₂ + c₁c₂) / [√(a₁² + b₁² + c₁²)√(a₂² + b₂² + c₂²)]|

• Now if l₁l₂ + m₁m₂ + n₁n₂ = 0, then the lines are perpendicular.
• If l₁ = l₂, m₁ = m₂, and n₁ = n₂, then lines are parallel.
• Similarly, if a₁a₂ + b₁b₂ + c₁c₂ = 0, then the lines are perpendicular.
• If a₁ / a₂, b₁ / b₂, and c₁ / c₂, then lines are parallel.

Equation of Straight Line Passing Through a Given Point

• Let a straight line pass through a point A with a position vector \(\vec{a}(x_1 \hat{i}+y_1\hat{j}+z_1\hat{k})\) and parallel to a vector \(b(a\hat{i}+b\hat{j}+c\hat{k})\), then its equation is given as
• In Vector Form:

\(\vec{r} = \vec{a} + \lambda \vec{b}\)

• In Cartesian Form:

(x - x₁)/a = (y - y₁)/b = (z - z₁)/c

Where a, b, and c are direction ratios.

A straight line passing through a point

Equation of Straight Line Passing Through Two Points

• In Vector Form: The equation of a line passing through two points whose position vectors are \(\vec{a}\) and \(\vec{b}\) is

\(\vec{r} = \vec{a} + \lambda (\vec{b} - \vec{a})\)

• In Cartesian Form: The equation of a straight line passing through (x₁, y₁, z₁) and (x₂, y₂, z₂) is

(x - x₁) / (x₂ - x₁) = (y - y₁) / (y₂ - y₁) = (z - z₁) / (z₂ - z₁)

A straight line passing through two points

Angle Between Two Lines

• In Vector Form: Let \(\vec{r} = \vec{a_1} + \lambda \vec{b_1} \) and \(\vec{r} = \vec{a_2} + \lambda \vec{b_2} \)be the equations of two straight lines. If θ is the angle between them, then

\(cos \theta = \frac{\vec{b_1} . \vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\)

• In Cartesian Form: Let (x - x₁)/a₁ = (y - y₁)/b₁ = (z - z₁)/c₁ and (x - x₂)/a₂ = (y - y₂)/b₂ = (z - z₂)/c₂ be the equations of two straight lines. If θ is the angle between them, then

cos θ = (a₁a₂ + b₁b₂ + c₁c₂) / [√(a₁² + b₁² + c₁²)√(a₂² + b₂² + c₂²)]

Shortest Distance Between Two Lines

• Let the straight lines are (x - x₁)/a₁ = (y - y₁)/b₁ = (z - z₁)/c₁ and (x - x₂)/a₂ = (y - y₂)/b₂ = (z - z₂)/c₂ and d is the shortest distance between them, then

d = |(x₁ – x₂)l + (y₁ – y₂)m + (z₁ – z₂)n|

Where l, m, and n are directional cosines.

• If \(\vec{r} = \vec{a_1} + \lambda \vec{b_1} \) and \(\vec{r} = \vec{a_2} + \lambda \vec{b_2} \) are two skew lines, then the distance between them is given by

\(|\frac{(\vec{b_1} \times \vec{b_2})(\vec{a_2}- \vec{a_1})}{|\vec{b_1 \times} |}|\)

There are Some important List Of Top Mathematics Questions On Three-Dimensional Geometry Asked In CBSE CLASS XII