Maharashtra State Board · Class 12 · Mathematics · Part I Chapter 6

Line and Plane — Formula Sheet

Board Formulas

16 formulas

1.Line through a Point, Parallel to a Vector
$r = a + λ b$
$a$ : Position vector of a known point A on the line · $b$ : Direction vector of the line · $λ$ : Scalar parameter (λ ∈ R)
Each value of the scalar λ gives one point on the line. Any non-zero multiple of b is also a direction vector.
2.Cartesian Form of a Line
$\frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = \frac{z - z _{1}}{c}$
$(x_{1}, y_{1}, z_{1})$ : A point on the line · $a, b, c$ : Direction ratios of the line
Symmetric form of the same line. Setting each ratio equal to λ gives the general point (x₁ + aλ, y₁ + bλ, z₁ + cλ).
3.Line through Two Points
$r = a + λ (b - a), \frac{x - x _{1}}{x _{2} - x _{1}} = \frac{y - y _{1}}{y _{2} - y _{1}} = \frac{z - z _{1}}{z _{2} - z _{1}}$
The direction vector is the vector joining the two points, b − a.
4.Angle between Two Lines
$cos θ = \frac{∣ b _{1} \cdot b _{2} ∣}{∣ b _{1} ∣ ∣ b _{2} ∣} = \frac{∣ a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2} ∣}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}$
Perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0. Parallel: a₁/a₂ = b₁/b₂ = c₁/c₂. Only the direction vectors matter.
5.Distance of a Point from a Line★
$d = \frac{∣ ( α - a ) \times b ∣}{∣ b ∣} = ∣ α - a ∣^{2} - [(α - a) \cdot b^]^{2}$
$α$ : Position vector of the given point P · $b^$ : Unit vector along the line
The second form is Pythagoras: AP² minus the square of the projection of AP on the line.
6.Shortest Distance between Skew Lines★
$d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}$
For lines r = a₁ + λb₁ and r = a₂ + μb₂ that are neither parallel nor intersecting. The numerator is a scalar triple product.
7.Distance between Parallel Lines
$d = \frac{∣ ( a _{2} - a _{1} ) \times b ∣}{∣ b ∣}$
For r = a₁ + λb and r = a₂ + μb. This is the distance of the point A₂ from the first line.
8.Condition for Two Lines to Intersect (Coplanarity)★
$(a_{2} - a_{1}) \cdot (b_{1} \times b_{2}) = 0 ⟺ x_{2} - x_{1} a_{1} a_{2} y_{2} - y_{1} b_{1} b_{2} z_{2} - z_{1} c_{1} c_{2} = 0$
Two non-parallel lines intersect exactly when they are coplanar, i.e. when the shortest distance is zero.
9.Plane in Normal Form
$r \cdot n^= p ⟺ l x + m y + n z = p$
$n^$ : Unit vector normal to the plane · $p$ : Distance of the plane from the origin
p ≥ 0 is the perpendicular distance of the plane from the origin and (l, m, n) are the direction cosines of the normal. To reduce ax + by + cz = d (d > 0), divide by √(a² + b² + c²).
10.Plane through a Point, Perpendicular to a Vector
$(r - a) \cdot n = 0 ⟺ a (x - x_{1}) + b (y - y_{1}) + c (z - z_{1}) = 0$
Equivalently r · n = a · n. Here (a, b, c) are direction ratios of the normal and (x₁, y₁, z₁) is the given point.
11.Plane through Three Non-collinear Points★
$x - x_{1} x_{2} - x_{1} x_{3} - x_{1} y - y_{1} y_{2} - y_{1} y_{3} - y_{1} z - z_{1} z_{2} - z_{1} z_{3} - z_{1} = 0$
Vector form: (r − a) · [(b − a) × (c − a)] = 0. The cross product (b − a) × (c − a) is a normal to the plane.
12.Plane through the Intersection of Two Planes
$r \cdot (n_{1} + λ n_{2}) = d_{1} + λ d_{2}$
Cartesian: (a₁x + b₁y + c₁z − d₁) + λ(a₂x + b₂y + c₂z − d₂) = 0. One more condition (a point, or parallel/perpendicular to something) fixes λ.
13.Angle between Two Planes
$cos θ = \frac{∣ n _{1} \cdot n _{2} ∣}{∣ n _{1} ∣ ∣ n _{2} ∣}$
The angle between planes is the angle between their normals. Perpendicular planes: n₁ · n₂ = 0; parallel planes: n₁ = kn₂.
14.Angle between a Line and a Plane
$sin θ = \frac{∣ b \cdot n ∣}{∣ b ∣ ∣ n ∣}$
$b$ : Direction vector of the line · $n$ : Normal vector of the plane
θ is the complement of the angle between the line and the normal, so sin replaces cos.
15.Line Parallel or Perpendicular to a Plane
$b \cdot n = 0 (line parallel to plane), b = k n (line perpendicular to plane)$
A line parallel to the plane is perpendicular to its normal; a line perpendicular to the plane is along its normal.
16.Distance of a Point from a Plane★
$d = \frac{∣ α \cdot n - p ∣}{∣ n ∣} = \frac{∣ a x _{1} + b y _{1} + c z _{1} + d _{0} ∣}{a ^{2} + b ^{2} + c ^{2}}$
$(x_{1}, y_{1}, z_{1})$ : The given point, with position vector α · $d_{0}$ : Constant term of the plane ax + by + cz + d₀ = 0
First form for the plane r · n = p; second for ax + by + cz + d₀ = 0. Move every term to the left side before substituting.