Maharashtra State Board · Class 12 · Mathematics · Part I Chapter 5

Vectors and Three-Dimensional Geometry — Formula Sheet

Board Formulas

12 formulas

1.Vector Addition and Scalar Multiplication
$a + b = (a_{1} + b_{1}) i^+ (a_{2} + b_{2}) j^+ (a_{3} + b_{3}) k^$
Component-wise addition. Scalar multiplication kâ = ka₁î + ka₂ĵ + ka₃k̂.
2.Magnitude and Unit Vector★
$∣ a ∣ = a_{1}^{2} + a_{2}^{2} + a_{3}^{2}, a^= \frac{a}{∣ a ∣}$
Magnitude is nonnegative. Unit vector has magnitude 1.
3.Dot Product★
$a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$
Scalar quantity. â·b̂ = cos θ; perpendicular ⟺ a·b = 0.
4.Cross Product★
$a \times b = i^a_{1} b_{1} j^a_{2} b_{2} k^a_{3} b_{3}$
Vector perpendicular to both a and b. |a × b| = |a||b|sin θ. Right-hand rule.
5.Area of Parallelogram / Triangle
$Parallelogram: ∣ a \times b ∣; Triangle: \frac{1}{2} ∣ a \times b ∣$
Simple geometric use of cross product.
6.Scalar Triple Product★
$[a b c] = a \cdot (b \times c) = a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3}$
Volume of parallelepiped. Cyclic permutation preserves value; swap changes sign.
7.Coplanarity Condition★
$[a b c] = 0$
Three vectors are coplanar iff their scalar triple product vanishes.
8.Vector Line Equation★
$r = a + λ b$
a = position vector of a point on line; b = direction vector; λ ∈ R.
9.Cartesian Line Equation
$\frac{x - x _{0}}{l} = \frac{y - y _{0}}{m} = \frac{z - z _{0}}{n}$
(l, m, n) direction ratios; (x₀, y₀, z₀) a point on line.
10.Plane in Vector Form★
$r \cdot n^= d$
n̂ = unit normal, d = perpendicular distance from origin. Cartesian: lx + my + nz = d.
11.Angle between Two Lines / Planes
$cos θ = \frac{b _{1} \cdot b _{2}}{∣ b _{1} ∣∣ b _{2} ∣} (lines)$
For planes use normals; for line and plane use sin θ = |n·b|/(|n||b|).
12.Shortest Distance between Skew Lines★
$d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}$
Only for skew lines (non-parallel, non-intersecting). Coplanar ⇒ numerator = 0.