Vector Algebra

What is vector algebra?

Vector algebra is the area of mathematics that focuses on algebraic operations involving vector quantities, such as addition, scalar multiplication, and related properties.

Fundamental Properties of Vector Algebra

Associative Property of Addition

For any three vectors, the sum is independent of the grouping of the terms:

$$ \vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c} $$

Commutative Property of Addition

The order of addition does not affect the result:

$$ \vec{a} + \vec{b} = \vec{b} + \vec{a} $$

Distributive Property over Scalar Addition

Given a vector and two scalars \( k \) and \( j \), the following holds:

$$ (k + j)\vec{a} = k \cdot \vec{a} + j \cdot \vec{a} $$

Distributive Property over Vector Addition

A scalar multiplying the sum of two vectors distributes over the addition:

$$ k(\vec{a} + \vec{b}) = k \cdot \vec{a} + k \cdot \vec{b} $$

Associative Property of Scalar Multiplication

Scalar multiplication is associative with respect to scalars:

$$ k(j \cdot \vec{a}) = (k \cdot j)\vec{a} $$

Existence of the Zero Vector

There exists a unique zero vector \( \vec{0} \) such that adding it to any vector leaves the vector unchanged:

$$ \vec{a} + \vec{0} = \vec{a} $$

Existence of Additive Inverses

For every vector, there exists an additive inverse that yields the zero vector when summed with the original:

$$ \vec{a} + (-\vec{a}) = \vec{0} $$

These properties form the foundation of vector spaces and are essential for understanding linear algebra and its applications in physics, engineering, and beyond.