We will use the cross product and dot product of vectors to explore equations of lines and planes in 3 dimensional space. Vector functions have an input
t and an output of a vector function of t.
A position vector is a vector whose initial point is fixed at the origin so that each point corresponds to
P = <x,y>. Since a position vector...
Vectors in Three Dimensional Space
In single variable calculus, or Calc 1 and 2, we have dealt with functions in two dimensions, or R2. In multivariable calculus, we will need to get accustomed to working in three dimensional space, or R3. Most of our notation and calculation will be the same, but with the extension of an added variable,
The extended Cartesian graph now looks...
Properties of Vectors
Vectors follow most of the same arithemetic rules as scalar numbers. The following are various properties that apply to vectors in two dimensional and three dimensional space and are important to keep in mind
Addition of Vectors
Scalar and Vector Properties
Dot Product Properties
The Dot Product is defined as
Vectors are usually used to represent velocity and acceleration, force, and other directional quantities in physics.
Vectors are quantities with size and direction.
The objects that we have worked with in single variable calculus (Calculus 1 and 2) have all had a quantity, i.e. we were able to measure them.
Some quantities only have size, such as time, temperature,...
In calculus, we have dealt with
functions of x in two dimensional space. Multivariable Calculus, also known as Vector Calculus, deals with functions of two variables in 3 dimensional space, as well as computing with vectors instead of lines.
In single variable calculus, we see that y is a function of x
In multivariable calculus, z is a function...
Calculus Help and Problems
This section contains in depth discussions and explanations on key topics that appear throughout Calculus 1 and 2 up through Vector Calculus. The topics are arranged in a natural progression catering typically to late highschool and early college students, covering the foundations of calculus, limits, derivatives, integrals, and vectors.
Still need help...