Learning Objectives
- Understand vectors, their types, and representations
- Learn vector addition, scalar multiplication, and components
- Study dot product (scalar product) and cross product (vector product)
- Apply vector algebra to geometric problems
Key Concepts
Basic Concepts
A vector has both magnitude and direction. Notation: a or a→. Magnitude: |a|. Unit vector: â = a/|a| (magnitude 1). Zero vector (0→): Magnitude 0, arbitrary direction. Position vector: Vector from origin to point P. If P = (x, y, z), then OP→ = xi + yj + zk. Equal vectors: Same magnitude and direction. Collinear vectors: Parallel (same or opposite direction); a = λb for some scalar λ.
Component form: a = a₁i + a₂j + a₃k. |a| = √(a₁2 + a₂2 + a₃2). Direction cosines: l = a₁/|a|, m = a₂/|a|, n = a₃/|a|. l2 + m2 + n2 = 1.
Vector Operations
Addition: a + b = (a₁+b₁)i + (a₂+b₂)j + (a₃+b₃)k. Triangle law, parallelogram law. Scalar multiplication: ka = ka₁i + ka₂j + ka₃k.
Section formula: Position vector of point dividing AB in ratio m:n = (mb + na)/(m + n).
Scalar (Dot) Product
a · b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃. Properties: Commutative (a·b = b·a). Distributive (a·(b+c) = a·b + a·c). i·i = j·j = k·k = 1; i·j = j·k = k·i = 0. a · b = 0 iff a ⊥ b (perpendicular). Projection of a on b = (a·b)/|b|. Angle between vectors: cosθ = (a·b)/(|a||b|).
Vector (Cross) Product
a × b = |a||b|sinθ n̂ where n̂ is unit vector perpendicular to both (right-hand rule). a × b = | i j k | (determinant with components). |a × b| = area of parallelogram with sides a, b. Properties: Not commutative: a × b = -(b × a). i × i = 0; i × j = k; j × k = i; k × i = j. a × b = 0 iff a ∥ b (parallel/collinear). Area of triangle = (1/2)|a × b|.
Scalar Triple Product
[a b c] = a · (b × c) = determinant of 3×3 matrix of components. Geometrically: Volume of parallelepiped = |[a b c]|. [a b c] = 0 iff vectors are coplanar. Properties: [a b c] = [b c a] = [c a b] (cyclic permutation preserves value); [a b c] = -[b a c] (swapping changes sign).
Summary
Vectors are quantities with magnitude and direction. The dot product gives a scalar and measures projection. The cross product gives a vector perpendicular to both inputs. The scalar triple product gives the volume of a parallelepiped and tests coplanarity.
Important Terms
- Dot product: a·b = |a||b|cosθ; scalar result
- Cross product: a×b = |a||b|sinθ n̂; vector result
- Direction cosines: Cosines of angles with coordinate axes; l2+m2+n2=1
- Scalar triple product: [a b c] = a·(b×c); volume of parallelepiped
- Coplanar: Vectors in the same plane; [a b c] = 0
Quick Revision
- |a| = √(a₁2+a₂2+a₃2); unit vector = a/|a|
- a·b = a₁b₁+a₂b₂+a₃b₃; a·b = 0 → perpendicular
- a×b: use determinant method; |a×b| = area of parallelogram
- a×b = 0 → parallel; a·b = 0 → perpendicular
- Volume of parallelepiped = |a·(b×c)| = |scalar triple product|
- Coplanar vectors: [a b c] = 0
- Projection of a on b = (a·b)/|b|