Chapter 10 — Vector Algebra

What is a vector? A vector is a quantity that has both magnitude (size) and direction. Unlike scalars (which have only magnitude), vectors represent directed quantities such as displacement, velocity, and force. Vectors are usually denoted by boldface letters (a) or letters with an arrow on top (\u2192a).

Representation: Graphically a vector is shown as a directed line segment (an arrow). Algebraically in coordinate form in 2D: \u2192a = (x, y) or as a combination of unit vectors i and j: \u2192a = x i + y j. In 3D: \u2192a = (x, y, z) = x i + y j + z k.

Key types and definitions

• Zero vector: The vector with zero magnitude and undefined direction, denoted \u21920 or (0,0,0).
• Unit vector: A vector of unit magnitude. The unit vector in the direction of \u2192a is \u2192a/|\u2192a|.
• Negative of a vector: The vector with same magnitude but opposite direction: -\u2192a.
• Position (or radius) vector: The vector from the origin O to a point P(x,y,z) is \u2192OP = (x,y,z).
• Collinear vectors: Vectors that lie along the same line (one is a scalar multiple of the other).

Equality of vectors: Two vectors are equal if they have the same magnitude and same direction. In components: (x1,y1,z1) = (x2,y2,z2) iff x1=x2, y1=y2, z1=z2.

Operations

• Addition: Graphically by the triangle (head-to-tail) or parallelogram rule. Algebraically add components: \u2192a + \u2192b = (ax+bx, ay+by, az+bz).
• Subtraction: \u2192a - \u2192b = \u2192a + (-\u2192b) (subtract components).
• Scalar multiplication: For scalar \u03bb, \u03bb\u2192a scales magnitude and possibly reverses direction if \u03bb<0: \u03bb\u2192a = (\u03bb ax, \u03bb ay, \u03bb az).

Important properties

• Commutative law: \u2192a + \u2192b = \u2192b + \u2192a.
• Associative law: (\u2192a + \u2192b) + \u2192c = \u2192a + (\u2192b + \u2192c).
• Distributive: \u03bb(\u2192a + \u2192b) = \u03bb\u2192a + \u03bb\u2192b.
• Scalar magnitude: |\u03bb\u2192a| = |\u03bb| |\u2192a|.
• Triangle inequality: |\u2192a + \u2192b| \u2264 |\u2192a| + |\u2192b|.

These basic concepts form the foundation for vector algebra used in geometry, physics and engineering problems.

What is a vector? A vector is a quantity that has both magnitude (size) and direction. Common examples: displacement, velocity, force.

Geometric representation: A vector is shown as a directed line segment (an arrow). The length of the arrow represents magnitude and the arrowhead shows direction. If the tail is at the origin and the head at point (x,y) (or (x,y,z)), the arrow represents the position vector of that point.

Algebraic (component) representation:

• In 2D: v = xi + yj, where x and y are the components along the x- and y-axes, and i, j are the unit vectors along those axes.
• In 3D: v = xi + yj + zk, where x, y, z are components along x, y, z axes and k is the unit vector along z.
• Also written in coordinate form: v = <x, y> (2D) or v = <x, y, z> (3D), or as a column: [x y]^T or [x y z]^T.

Components from two points: If A(x1,y1) and B(x2,y2) are two points, the vector from A to B is AB = <x2 - x1, y2 - y1>. Similarly for 3D: AB = <x2 - x1, y2 - y1, z2 - z1>.

Magnitude: For v = <x, y> the magnitude |v| = sqrt(x^2 + y^2). For v = <x, y, z>: |v| = sqrt(x^2 + y^2 + z^2).

Direction:

• In 2D a vector's direction can be given by the angle θ it makes with the positive x-axis: v = |v|(cosθ i + sinθ j), so components are x = |v| cosθ, y = |v| sinθ.
• In 3D direction cosines: if v = <x,y,z> and |v| = r, then cosα = x/r, cosβ = y/r, cosγ = z/r, where α, β, γ are angles with x, y, z axes.

Unit vector: The unit vector in direction of v is v̂ = v / |v| = <x/|v|, y/|v|, z/|v|>.

Important properties:

• Two vectors are equal iff their corresponding components are equal.
• The zero vector has all components zero and no defined direction.
• Scalar multiplication scales each component: c v = <c x, c y, c z>.

This component representation makes calculations (addition, subtraction, dot/cross products, magnitudes, projections) straightforward and connects geometric intuition with algebraic operations.

Overview: Vectors are quantities with magnitude and direction. Addition and subtraction combine such quantities either graphically (geometric rules) or algebraically (component-wise). These operations give the resultant vector.

Graphical methods:

• Triangle (tip-to-tail) rule: Place the tail of vector b at the tip of vector a; the vector from the tail of a to the tip of b is a + b.
• Parallelogram rule: Place both vectors with a common origin; complete the parallelogram. The diagonal from the origin is a + b.
• Subtraction: a − b = a + (−b). Reverse b (change its direction) and add by the rules above.

Algebraic (component) method: For vectors in 2D, write a = (a_x, a_y), b = (b_x, b_y). Then

• a + b = (a_x + b_x, a_y + b_y)
• a − b = (a_x − b_x, a_y − b_y)

Magnitude and direction of the resultant: If θ is the angle between a and b, then

• |a + b|^2 = |a|^2 + |b|^2 + 2|a||b| cosθ (law of cosines)
• |a − b|^2 = |a|^2 + |b|^2 − 2|a||b| cosθ
• If a + b = (r_x, r_y), its direction φ (measured from x-axis) is φ = arctan(r_y / r_x) (taking quadrant into account).

Properties: Addition of vectors is commutative and associative: a + b = b + a, (a + b) + c = a + (b + c). The zero vector 0 is the additive identity: a + 0 = a. Every vector a has an additive inverse −a: a + (−a) = 0.

Notes for 3D: Same rules apply with components (a_x, a_y, a_z). Addition and subtraction are performed component-wise.

Definition: Multiplication of a vector by a scalar means scaling the vector by a real number. If a is a vector and k is a scalar (real number), the product k a is a vector obtained by multiplying each component of a by k.

Algebraic form: If a = (x, y) in 2D or a = (x, y, z) in 3D, then

• k a = (k x, k y) in 2D
• k a = (k x, k y, k z) in 3D

Geometric effect:

• Magnitude: |k a| = |k| · |a|. The length of the vector is multiplied by |k|.
• Direction: If k > 0, k a has the same direction as a. If k < 0, k a has the opposite direction (it is reversed). If k = 0, k a is the zero vector.
• Collinearity: k a lies on the line through the origin and the head of a (i.e., k a is collinear with a).

Unit vector relation: If a ≠ 0 and û = a/|a| is the unit vector in the direction of a, then k a = (k |a|) û. This shows scaling acts only on magnitude.

Key vector-scalar properties:

• Associativity of scalars: (k₁ k₂) a = k₁ (k₂ a).
• Distributivity over vector addition: k (a + b) = k a + k b.
• Distributivity over scalar addition: (k₁ + k₂) a = k₁ a + k₂ a.
• 1 · a = a and 0 · a = 0 (zero vector).

Visual intuition: Draw vector a from the origin to point P(x,y). For k > 1 the arrow extends beyond P in the same line; for 0 < k < 1 the arrow is shorter and lies between origin and P; for k < 0 the arrow points in the exact opposite direction on the same line; for k = 0 the arrow collapses to the origin.

Definition: The dot (or scalar) product is a binary operation on two vectors that returns a scalar. For vectors a and b, their dot product is written a · b.

Algebraic form (coordinates): If a = (a1, a2, ..., an) and b = (b1, b2, ..., bn) in R^n, then a · b = a1 b1 + a2 b2 + ... + an bn.

Geometric form: If |a| and |b| are magnitudes and θ is the angle between a and b (0 ≤ θ ≤ π), then

a · b = |a| |b| cos θ.

These two expressions are equivalent (can be shown using the law of cosines). The dot product therefore connects components and geometry.

Important consequences and interpretations:

• a · a = |a|^2, so the magnitude of a is |a| = sqrt(a · a).
• Two nonzero vectors are orthogonal (perpendicular) iff a · b = 0.
• The scalar projection (component) of a on b: comp_b(a) = (a · b)/|b|. The vector projection of a on b: proj_b(a) = ((a · b)/|b|^2) b.
• Angle between vectors: cos θ = (a · b)/(|a||b|).
• Distributive and bilinear properties: a · (b + c) = a · b + a · c and (λa) · b = λ (a · b).

Physical meaning and applications: The dot product measures how much one vector goes in the direction of another. In physics, work done by a force F during displacement d is W = F · d = |F||d|cos θ. Dot products appear in projections, orthogonality tests, computing angles, least-squares, and similarity measures (cosine similarity) in data science.

Useful inequalities: Cauchy–Schwarz: |a · b| ≤ |a| |b|, which yields the triangle inequality for lengths.

Definition: For two vectors a and b in three-dimensional space, the cross (vector) product a × b is a vector perpendicular to both a and b. Its direction is given by the right-hand rule and its magnitude is |a||b|sinθ, where θ is the smaller angle between a and b (0 ≤ θ ≤ π).

Computation (components): If a = (a1, a2, a3) and b = (b1, b2, b3), then

a × b = (a2 b3 − a3 b2, a3 b1 − a1 b3, a1 b2 − a2 b1)

Equivalently, a × b can be written as the determinant of a symbolic matrix: a × b = det([i j k; a1 a2 a3; b1 b2 b3]).

Direction (right-hand rule): Point your index finger along a and middle finger along b; your thumb then points in the direction of a × b. If you reverse the order, b × a = −(a × b).

Geometric meaning: The magnitude |a × b| equals the area of the parallelogram spanned by a and b. The area of the triangle with sides a and b from the same vertex is (1/2)|a × b|.

When it is zero: a × b = 0 exactly when a and b are parallel (θ = 0 or π) or when one vector is the zero vector.

Common identities: distributive over addition: a × (b + c) = a × b + a × c; anti-commutative: a × b = −(b × a); scalar triple product relation: a · (b × c) equals the volume of the parallelepiped formed by a, b, c.

Applications (brief): computing normals to planes, torque (τ = r × F), magnetic force on a moving charge (F = q v × B), angular momentum (L = r × p), and areas in geometry.

The scalar triple product (also called the box product) of three vectors u, v and w is defined as u · (v × w). It is a scalar whose absolute value equals the volume of the parallelepiped formed by the three vectors when placed tail-to-tail, and whose sign indicates the orientation (right-handed or left-handed) of the ordered triple (u,v,w).

Algebraic definition and determinant form:

• u · (v × w) = (u × v) · w.
• If u = (u1,u2,u3), v = (v1,v2,v3), w = (w1,w2,w3), then the scalar triple product equals the determinant of the 3x3 matrix with rows (or columns) u, v, w: u · (v × w) = det [[u1,u2,u3],[v1,v2,v3],[w1,w2,w3]].

Geometric meaning:

• |u · (v × w)| = volume of the parallelepiped with edges u, v, w.
• Volume of the tetrahedron formed by these three edge-vectors = |u · (v × w)| / 6 (parallelepiped has 6 congruent tetrahedra).
• If u · (v × w) = 0, the vectors are coplanar (volume zero).
• The sign of u · (v × w) is positive if (u,v,w) is a right-handed set and negative if left-handed; this follows from the right-hand rule for the cross product.

Key algebraic properties (useful for calculation):

• Distributive: u · (v + w) × z = u · (v × z) + u · (w × z).
• Scalar factor: (a u) · (v × w) = a [u · (v × w)].
• Permutation: u · (v × w) = v · (w × u) = w · (u × v) (cyclic permutations keep the same value).
• Swap sign: swapping any two vectors changes the sign: u · (v × w) = -v · (u × w).

Computation: Expand the determinant or compute v × w first and then take the dot product with u. For numeric vectors use the standard 3x3 determinant expansion or Sarrus' rule.

Overview
There are two related concepts called the vector triple product and the scalar (or mixed) triple product. Both are important in geometry and physics (volumes, torque, moments).

1. Scalar (mixed) triple product
Definition: A · (B × C). It is a scalar equal to the signed volume of the parallelepiped formed by A, B and C. In coordinates it equals the determinant whose rows (or columns) are the components of A, B, C.

• Geometric meaning: |A · (B × C)| = volume of parallelepiped with edges A, B, C.
• Coplanarity test: A · (B × C) = 0 ⇔ A, B, C are coplanar.
• Permutation property: A · (B × C) = B · (C × A) = C · (A × B) (cyclic). Swapping two vectors changes sign.

2. Vector triple product
Definition: A × (B × C) is a vector. The key identity (called BAC–CAB or Lagrange's identity) is:

A × (B × C) = B (A · C) − C (A · B)

This identity expresses the result as a linear combination of B and C (i.e. it lies in the plane spanned by B and C). A similar identity for the other ordering is:

(A × B) × C = B (C · A) − A (B · C)

Short proof idea of A × (B × C) = B(A·C) − C(A·B)
Decompose A into two parts: A = A_parallel + A_perp, where A_parallel lies in the plane spanned by B and C, and A_perp is perpendicular to that plane. Since A_perp is parallel to B × C, A_perp × (B × C) = 0. Hence only A_parallel matters. Write A_parallel = uB + vC and find u, v by dotting with C and B respectively (using A·B and A·C). Substituting gives the identity above.

Useful consequences

• A × (B × C) lies in span{B, C} (it is orthogonal to B × C).
• Scalar triple product equals determinant: A · (B × C) = det([A B C]).
• If any two of A, B, C are equal or linearly dependent, the scalar triple product is zero.

Applications / significance
- Volume computation of parallelepipeds and tetrahedra (via scalar triple product).
- Determining coplanarity of vectors.
- Simplifying expressions in mechanics and electromagnetism (e.g., manipulations of torque τ = r × F and identities in angular momentum and Lorentz force algebra).
- Simplifying cross-product algebra in rigid-body kinematics and vector calculus.

Overview
Geometrical applications of vectors use vector operations (addition, scalar multiplication, dot and cross products) to describe and solve problems about points, lines, planes, distances, angles, areas and volumes in 2D and 3D space. Vectors provide compact coordinate-free formulas that are especially useful in analytic geometry and physics.

Basic ideas

• Position vector: The vector from the origin O to point P is r = OP.
• Vector equation of a line: A line through point A (position vector a) with direction vector b: r = a + t b, t ∈ R.
• Vector equation of a plane: A plane with normal vector n and at distance d from origin: r · n = d. Or through point A (position a): (r − a) · n = 0.
• Direction cosines: For a direction vector v = (l, m, n), the direction cosines are proportional to its components.

Key geometric operations

• Angle between two vectors b1 and b2: cos θ = (b1 · b2) / (|b1||b2|).
• Projection of a on b (scalar): comp_b(a) = (a · b) / |b|. (vector projection) proj_b(a) = ((a · b) / |b|^2) b.
• Area of triangle/parallelogram: If two sides are represented by vectors u and v, area of parallelogram = |u × v|, area of triangle = 1/2 |u × v|.
• Volume: Volume of parallelepiped formed by vectors a, b, c = |a · (b × c)|. Volume of corresponding tetrahedron = (1/6) |a · (b × c)|.
• Coplanarity: Points (or vectors) a, b, c are coplanar iff scalar triple product a · (b × c) = 0.

Distances

• Distance from point P (position p) to line L: r = a + t b is |(p − a) × b| / |b| (length of perpendicular).
• Distance from point P (p) to plane (r · n = d): |p · n − d| / |n|.
• Shortest distance between two skew lines r = a1 + λ b1 and r = a2 + μ b2:
  distance = |(a2 − a1) · (b1 × b2)| / |b1 × b2|.

How to use these in typical problems

• To check if two lines intersect: equate r = a1 + λ b1 and r = a2 + μ b2 and solve for λ, μ. If consistent, they intersect; otherwise they are skew or parallel.
• To find foot of perpendicular from P to line or plane: use projection formulas or solve for parameter t by minimizing distance (set derivative or use orthogonality).
• To get angle between line and plane: get angle between line direction b and plane normal n; angle between line and plane = 90° − angle(b, n).

Why vectors help
Vector formulas replace coordinate-heavy algebra by compact algebraic operations (dot and cross products) that are easy to compute and generalize to 3D problems in physics (forces, velocities), engineering (structural geometry), and computer graphics (rendering, collision detection).