The Vector

Definition

A vector has direction and magnitude.

Remark

Well, technically, anything that fulfils the criteria of a vector space is a vector.

Vector in Component Form

Definition

is the set of all vectors with components, each of which is a real number.

Vector Visualization.png

Vector Equality

Definition

It all the components of and are equal, then

Zero Vector

Definition

is a zero vector, and all it's components are

Vector Addition

Definition

Vectors can be added by adding their components

Scalar Multiplication

Definition

Vector components are all multiplied by a scalar value

Parallel Vectors

Definition

Two nonzero vectors are parallel if they are scalar multiples of each other

Two lines in are parallel if their direction Vectors are parallel. Two Planes in are parallel if their normal vectors are parallel

Linear Combination

Definition

Two or more vectors whose respective components are multiplied, and then added with each other.

Let and for some positive integer

is a linear combination of the vectors .

Visualization