Math Flashcards
The zero Vector
- Every point in the x, y plane and the x, y, z plane has an associated vector ending at that point
- Even the point (0,0) or (0,0,0)
this is called the zero vector
Two Vectors are the same if?
(1) They are the same dimension
(2) Thier components are all the same (e.g. same x component, same y component, ect)
Collinear
- if two vectors are equal to each other, they must necessarily be “collinear”, which means they lie on the same line.
- However, two non-equal vectors can still be collinear
Distance Between Vectors
- Yes, our Vectors are all thouching at the orgin
- By distance, we mean the distance between the end points of each vector
Pythagorean Theorem
Length of a Vector
- The length of a vector is defined as the distance formm where it starts (the origin) to where it ends (the endpoint)
- Other names for lenght of a vector include:
Magnitude, Norm
Unit Vectors
if a vector has a norm (length) equal to exactly 1, we call it a unit vector
Scalars
- A scalar is just a number (every element of R^1)
Scalar Multiplication
- Vector can be multiplied (“scaled”) by a scalar
- This “scalar multiplicaton” effects each component of the vector
Scalar Multiplication
- A scalar multiple of a vector will always be collinear (parallel) to that vector
- Also, by definition, the zero vector is collinear to every vector
True or False
Vectors u and v are collinear if and only if there is some scalar value c such that v = cu
TRUE!!
Translation of Vectors
- A vector is a “Directed Line Segment” which starts at the origin.
- A Directed Line Segment, called PQ, form some point P to another point Q, where P is not the origin, is not a vector
Equivalent:
Two directed line segments which have the same magnitude and direction are called equivalent.
Midpoints of Vectors
The midpoint of a vector is the point half-way between the tip and the tail of the vector
Addition of Vectors
- Vectors can be added togthere just like how we add numbers togethere
must have the same number of components (R^2, R^3)
Negative of a Vector
the vector has the same magnitude of the vector but has an oppsite direction to the non negative vector
