1. Vectors Fundamentals: Components, Magnitude & Unit Vector
A vector in 3D space with components along Cartesian axes: a = a1 î + a2 ĵ + a3 k̂. Magnitude (length): |a| = √(a1² + a2² + a3²). Unit vector in the direction of a: â = a / |a| = (a1 î + a2 ĵ + a3 k̂) / √(a1² + a2² + a3²). Position vector of point P(x, y, z) from origin: r = x î + y ĵ + z k̂. Vector joining point A(x1, y1, z1) to B(x2, y2, z2): AB = (x2 - x1)î + (y2 - y1)ĵ + (z2 - z1)k̂.
2. Scalar (Dot) Product: Properties, Projections & Angle
Definition: a · b = |a| |b| cos θ = a1 b1 + a2 b2 + a3 b3. Angle between vectors: cos θ = (a · b) / (|a| |b|). Condition for perpendicularity: a · b = 0. Self dot product: a · a = |a|². Projection of vector a on vector b: Projection = (a · b) / |b|. Projection vector of a along b: Proj_b(a) = [(a · b) / |b|²] b. Commutative property: a · b = b · a.
3. Vector (Cross) Product: Magnitude, Normal Vector & Area
Definition: a × b = (|a| |b| sin θ) n̂, where n̂ is unit normal vector according to right-hand thumb rule. In determinant form: a × b = |[î, ĵ, k̂], [a1, a2, a3], [b1, b2, b3]|. Condition for parallelism: a × b = 0 (components are proportional: a1/b1 = a2/b2 = a3/b3). Non-commutative (anti-symmetric): a × b = -(b × a). Area of parallelogram with adjacent sides a and b: Area = |a × b|. Area of triangle with adjacent sides a and b: Area = (1/2) |a × b|.
4. Direction Cosines & Direction Ratios in 3D Space
If a directed line makes angles α, β, γ with positive X, Y, Z axes: Direction Cosines (DC): l = cos α, m = cos β, n = cos γ. Fundamental relation: l² + m² + n² = 1 (or cos² α + cos² β + cos² γ = 1, leading to sin² α + sin² β + sin² γ = 2). Direction Ratios (DR): Numbers a, b, c proportional to direction cosines: l = a / √(a² + b² + c²), m = b / √(a² + b² + c²), n = c / √(a² + b² + c²). Angle between two lines with DRs (a1, b1, c1) and (a2, b2, c2): cos θ = |a1 a2 + b1 b2 + c1 c2| / [√(a1² + b1² + c1²) √(a2² + b2² + c2²)]. Perpendicular condition: a1 a2 + b1 b2 + c1 c2 = 0.
5. Equations of a Straight Line in 3D Space
Line passing through point with position vector a and parallel to vector b: Vector Form: r = a + λ b. Cartesian Form: (x - x1) / a = (y - y1) / b = (z - z1) / c (where (x1, y1, z1) is the point and (a, b, c) are direction ratios). Line passing through two points (x1, y1, z1) and (x2, y2, z2): (x - x1) / (x2 - x1) = (y - y1) / (y2 - y1) = (z - z1) / (z2 - z1).
6. Shortest Distance Between Two Skew Lines
Skew lines are non-parallel, non-intersecting lines in 3D space: Line 1: r = a1 + λ b1; Line 2: r = a2 + μ b2. Shortest Distance d: d = |(b1 × b2) · (a2 - a1)| / |b1 × b2|. If the lines intersect, shortest distance d = 0, meaning (b1 × b2) · (a2 - a1) = 0. For parallel lines r = a1 + λ b and r = a2 + μ b: Shortest Distance d = |b × (a2 - a1)| / |b|.