CBSE · Class 12 · Mathematics · Chapter 11

Three Dimensional Geometry — Formula Sheet

Board Formulas

17 formulas

1.Direction Cosines
$l = cos α, m = cos β, n = cos γ$
$α, β, γ$ : Angles with the positive x, y and z-axes · $l, m, n$ : Direction cosines (pure numbers)
α, β, γ are the angles a directed line makes with the positive x, y and z-axes. Reversing the direction changes the sign of all three.
2.Identity for Direction Cosines
$l^{2} + m^{2} + n^{2} = 1$
True for every line. Use it to find a missing direction cosine, or to check your answer.
3.Direction Cosines from Direction Ratios★
$l = \pm \frac{a}{a ^{2} + b ^{2} + c ^{2}}, m = \pm \frac{b}{a ^{2} + b ^{2} + c ^{2}}, n = \pm \frac{c}{a ^{2} + b ^{2} + c ^{2}}$
$a, b, c$ : Direction ratios of the line
Direction ratios a, b, c are any numbers proportional to l, m, n. Take the same sign (all + or all −) in all three.
4.Direction Ratios of a Line Through Two Points
$x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1}$
For the line through P(x₁, y₁, z₁) and Q(x₂, y₂, z₂). Subtracting the other way round, x₁ − x₂ and so on, is also valid.
5.Direction Cosines of a Line Through Two Points
$\frac{x _{2} - x _{1}}{P Q}, \frac{y _{2} - y _{1}}{P Q}, \frac{z _{2} - z _{1}}{P Q}, P Q = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$
$P Q$ : Distance between the two points (units)
Divide each direction ratio by the distance PQ.
6.Collinearity of Three Points
$A, B, C collinear \Leftrightarrow d.r.s of A B and B C are proportional$
Proportional direction ratios make AB parallel to BC, and B is common to both, so the three points lie on one line.
7.Vector Equation of a Line
$r = a + λ b$
$a$ : Position vector of a known point on the line · $b$ : Vector parallel to the line (its components are direction ratios) · $λ$ : Real parameter
The line through the point with position vector a, parallel to b. Each real value of λ gives one point of the line.
8.Cartesian Equation of a Line★
$\frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = \frac{z - z _{1}}{c}$
The line through (x₁, y₁, z₁) with direction ratios a, b, c. The coefficient of x, y and z must be +1 in each numerator.
9.General Point on a Line
$(x_{1} + λa, y_{1} + λb, z_{1} + λ c)$
Parametric form. Use it when you need a point on the line that satisfies an extra condition.
10.Line Through Two Points
$r = a + λ (b - a), \frac{x - x _{1}}{x _{2} - x _{1}} = \frac{y - y _{1}}{y _{2} - y _{1}} = \frac{z - z _{1}}{z _{2} - z _{1}}$
This is the point–direction form with the direction taken as the vector from the first point to the second.
11.Angle Between Two Lines (Vector Form)
$cos θ = \frac{∣ b _{1} \cdot b _{2} ∣}{∣ b _{1} ∣ ∣ b _{2} ∣}$
For the lines r = a₁ + λb₁ and r = a₂ + μb₂. The modulus gives the acute angle.
12.Angle Between Two Lines (Direction Ratios)★
$cos θ = \frac{∣ a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2} ∣}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}$
The Cartesian form of the same result. Read the direction ratios from the denominators in standard form.
13.Angle Between Two Lines (Direction Cosines)
$cos θ = ∣ l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} ∣$
No denominator is needed, because l² + m² + n² = 1 for each line.
14.Perpendicular and Parallel Lines★
$⊥: a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0; ∥: \frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}$
Perpendicular: the dot product of the direction vectors is zero. Parallel: the direction ratios are proportional.
15.Shortest Distance Between Skew Lines (Vector Form)★
$d = \frac{( b _{1} \times b _{2} ) \cdot ( a _{2} - a _{1} )}{∣ b _{1} \times b _{2} ∣}$
$a_{1}, a_{2}$ : Position vectors of known points on the two lines · $b_{1}, b_{2}$ : Direction vectors of the two lines
Skew lines are neither parallel nor intersecting. The shortest distance is the projection of a₂ − a₁ on the common perpendicular b₁ × b₂.
16.Shortest Distance (Cartesian Form)
$d = \frac{x _{2} - x _{1} a _{1} a _{2} y _{2} - y _{1} b _{1} b _{2} z _{2} - z _{1} c _{1} c _{2}}{( b _{1} c _{2} - b _{2} c _{1} ) ^{2} + ( c _{1} a _{2} - c _{2} a _{1} ) ^{2} + ( a _{1} b _{2} - a _{2} b _{1} ) ^{2}}$
Same result as the vector form. The numerator is the scalar triple product and the denominator is |b₁ × b₂|.
17.Distance Between Parallel Lines★
$d = \frac{∣ b \times ( a _{2} - a _{1} ) ∣}{∣ b ∣}$
For r = a₁ + λb and r = a₂ + μb with the same direction b. If the second direction is a multiple of the first (for example 2b), use b.