, Definitions
Vector
Quantity that is characterised by both magnitude and
direction, and any object lives in a vector space.
Scalar
Quantity that is only characterised by magnitude, and does
not depend on the orientation of the coordinate axes.
Translation Invariance
Geometric property to be unaffected by a shift of the origin
of a coordinate system.
Vector Field
A vector that does depend on where in space it is, and has
components that are functions of position, such as electric
field at a point.
Isotropic
Looks the same in any direction we look.
Vector Space
, Vectors
Vector Representation
Geometrically, as an arrow in space.
Algebraically, in terms of its components. (𝐴𝐴𝑥𝑥 , 𝐴𝐴𝑦𝑦 ) are the
projections of 𝐴𝐴 onto the 𝑥𝑥- and 𝑦𝑦-axes.
�����⃗
A vector connecting two points: 𝐴𝐴𝐴𝐴
Length equal to the length of the line segment [𝐴𝐴𝐴𝐴].
- Vector Equality
�⃗ = 𝑊𝑊
Vectors with the same length and direction: 𝑉𝑉 ���⃗ .
- Vector Operations
Scalar Multiplication (shrinks/extends)
Vector Addition
The Triangle Rule
, Vector Addition is commutative:
Translation Invariance
Geometric property to be unaffected by a shift of the origin
of a coordinate system.
Vector
Quantity that is characterised by both magnitude and
direction, and any object lives in a vector space.
Scalar
Quantity that is only characterised by magnitude, and does
not depend on the orientation of the coordinate axes.
Translation Invariance
Geometric property to be unaffected by a shift of the origin
of a coordinate system.
Vector Field
A vector that does depend on where in space it is, and has
components that are functions of position, such as electric
field at a point.
Isotropic
Looks the same in any direction we look.
Vector Space
, Vectors
Vector Representation
Geometrically, as an arrow in space.
Algebraically, in terms of its components. (𝐴𝐴𝑥𝑥 , 𝐴𝐴𝑦𝑦 ) are the
projections of 𝐴𝐴 onto the 𝑥𝑥- and 𝑦𝑦-axes.
�����⃗
A vector connecting two points: 𝐴𝐴𝐴𝐴
Length equal to the length of the line segment [𝐴𝐴𝐴𝐴].
- Vector Equality
�⃗ = 𝑊𝑊
Vectors with the same length and direction: 𝑉𝑉 ���⃗ .
- Vector Operations
Scalar Multiplication (shrinks/extends)
Vector Addition
The Triangle Rule
, Vector Addition is commutative:
Translation Invariance
Geometric property to be unaffected by a shift of the origin
of a coordinate system.
