Vector algebra

In mathematics, vector algebra may mean:

• The operations of vector addition and scalar multiplication of a vector space
• The algebraic operations in vector calculus (vector analysis) – including the dot and cross products of 3-dimensional Euclidean space
• Algebra over a field – a vector space equipped with a bilinear product
• Any of the original vector algebras of the nineteenth century, including
  • Quaternions
  • Tessarines
  • Coquaternions
  • Biquaternions
  • Hyperbolic quaternions

1. Introduction to Vectors

• A vector is a quantity that has both magnitude and direction.
• Represented as A = (a₁, a₂, a₃) in 3D space.
• Notation: Bold letters (A), A with an arrow (), or component form.

2. Types of Vectors

• Zero Vector (\mathbf{0}): Magnitude is zero.

• Unit Vector (\hat{A}): Magnitude is one.

• Equal Vectors: Same magnitude and direction.

• Negative Vector: Same magnitude but opposite direction.

• Collinear Vectors: Parallel or anti-parallel vectors.

• Coplanar Vectors: Lie in the same plane.

3. Operations on Vectors

(i) Addition of Vectors

• Triangle Law: \mathbf{A} + \mathbf{B} = \mathbf{C}, placing the tail of \mathbf{B} at the head of \mathbf{A}.

• Parallelogram Law: The diagonal of the parallelogram represents the sum.

(ii) Subtraction of Vectors

• \mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B}

(iii) Scalar Multiplication

• k\mathbf{A}  scales the magnitude by  k  while maintaining direction.)

4. Components of a Vector

• In 3D Cartesian system:

\mathbf{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}

where \hat{i}, \hat{j}, \hat{k} are unit vectors along x, y, z axes.

• Magnitude:

|\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}

5. Scalar (Dot) Product

• \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos\theta

• Properties:

• Commutative: \mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}

• Distributive: \mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}

• \mathbf{A} \cdot \mathbf{A} = |\mathbf{A}|^2

6. Vector (Cross) Product

• \mathbf{A} \times \mathbf{B} = |\mathbf{A}||\mathbf{B}|\sin\theta \ \hat{n}

(Perpendicular to both \mathbf{A} and \mathbf{B})

• Properties:

• Anti-commutative: \mathbf{A} \times \mathbf{B} = - (\mathbf{B} \times \mathbf{A})

• Distributive: \mathbf{A} \times (\mathbf{B} + \mathbf{C}) = \mathbf{A} \times \mathbf{B} + \mathbf{A} \times \mathbf{C}

7. Triple Products

• Scalar Triple Product: \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})

Represents the volume of a parallelepiped.

• Vector Triple Product: \mathbf{A} \times (\mathbf{B} \times \mathbf{C})

Follows the BAC-CAB rule:

\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (\mathbf{A} \cdot \mathbf{C}) \mathbf{B} - (\mathbf{A} \cdot \mathbf{B}) \mathbf{C}

8. Applications of Vector Algebra

• Physics: Work (\mathbf{F} \cdot \mathbf{d}), Torque (\mathbf{r} \times \mathbf{F}), Motion.

• Engineering: Forces, Equilibrium, 3D modeling

• Computer Graphics: 3D transformations, animations.