Continuation of 12.1
Components are written with angle brackets to emphasize that they are vectors and not points.
Vectors have length, points do not.
Length of a Vector
length/magnitude/norm of
is defined by
Compute the distance from P to Q
Vector operations
Vector Addition
Scalar Multiplication
Computing Linear Combinations
Properties of vectors
Commutative
Associative
Distributive
Linear Combination
Linear Combination
A linear combination of
and is a vector of the form where are scalars
e.g.
Standard basis vectors
Anyis a unique linear combination of .
has the unique solution
Check:
Direction of vectors
Unit Vector
A vector of length 1 is called a unit vector.
is a unit vector iff
is a unit vector iff
Unit vectors
is a unit vector
is not a unit vector
If is a unit vector,
For any vector, we can obtain a unique unit vector in the same direction.
Check:
compute
*Final solution need not be simplified unless specified because calculators are not allowed on the exams.
12.2 - 3D Space: Surfaces, vectors, and curves
Right Hand Rule
The z axis is perpendicular to both the x and y axes. The conventional direction of z is determined by the right hand rule
Point your fingers extended outwards towards x, curl them towards y, the thumb is pointed to z.
3D Set Notation
*Set notation:
3D point and vector notation
Coordinate Planes
xy-plane=
xz-plane=
yz-plane=
3D vector operations are similar to those in 2D