Vectors Flashcards
Definition of a Vector
A vector is data of a number (or magnitude) and a direction
Definition The length of a vector
The length of a vector is called its magnitude or norm. the length of a vector is never negative
Caution with Vectors and Points
We have to distinguish vectors from points. Two points in a given order uniquely define a vector. But a vector does not uniquely define the points
zero vector
If the initial point coincides with the terminal point of a vector, then the length of the vector is zero. he vector with norm zero is called zero vector.
opposite vector
The vector with same norm but with opposite direction (in a representative, initial and terminal points are reversed) is called the opposite vector
Addition of vectors Definition
Let a and b two vectors. Choose representatives a and b so that the initial point of b is equal to the terminal point of a.
Then , the sum a+b is represented by the arrow connecting the initial point of a with the terminal point of b.
parallelogram-rule
a+b = b+a
Subtraction of vectors
The difference a - b of two vectros a and b is the sum of a and the opposite of b: a-b = a+(-b)
Multiplication of a vector with a scalar Definition
let a be a vector and r a scalar. The vector ra is the vector with magnitude |r| * |a| and has
-same direction as a if r >0
-opposite direction as a if r <0
Collinearity
If a and b are two vectors that have the same or opposite direction, then one of them is some multiple of the other
Collinearity definition
Two vectors a and b are called collinear or linearly dependent if
b = ra or
sb = a
Collinearity definition elegant
ra + sb = 0
position vector or location vector
We call OP the position vector or location vector of the point P. This means that a position vector always starts at the origin 0
Components of a vector connecting two points
a=PQ=OQ-OP
Norm of a vector
We calculate the norm of a vector by using Pythagorean Theorem
