Vectors
A vector is a quantity that has both magnitude and direction.
We are familiar with our 2-Dimensional Cartesian Plane but let us take a look at our 3 dimensional
Cartesian Plane.
In our 3 dimensional plane we have 3 axes, the x-axis, y-axis and z-axis. As such when we state a vector
in this plane we must state it using a 3 co-ordinate system.
In the above diagram we have the vector
Since our plane is defined in the form then the vector has the co-ordinates .
Unit Vectors
A unit vector is a vector that has a magnitude of 1 unit and acts in the direction of the vector. We can
define the unit vectors such that, represents a unit vector in the direction of the x-axis, as a unit vector
in the y-axis and as a unit vector in the z-axis.
As such, a vector , can be written in multiple forms. We can define in polar form as or vector form as
Operations on Vectors
Addition
When we add 2 vectors we add the corresponding coefficients of the unit vectors
given the vectors then
Subtraction
When we subtract 2 vectors we subtract the corresponding coefficients of the unit vectors
given the vectors then
Multiplication by a Scalar
When we do scalar multiplication we multiply the coefficients of the unit vectors by the scalar.
given the vector and the scalar quantity , then
Equality
Two vectors are said to be equal if the values of the corresponding unit vectors are equal.
given the vectors then
Types of Vectors
Position Vector
A position vector is a vector that has a fixed location on our Cartesian Plane. In general position vectors
have a fixed position in relation to the Origin. Therefore given the point , we can define the position
vector
Unit Vector
A unit vector is a vector that has a magnitude of 1 unit and acts in the direction of the original vector.
Since parallel vectors move/act in the same direction, we can use this knowledge to derive unit vectors.
In general 2 vectors are said to be parallel if we can represent one vector as a multiple of the other.
i.e. given the vectors then
(where is a scalar)
Simply put
Displacement Vector
A displacement vector is a vector whose position is not fixed in space. In general displacement vectors
are vectors that lie between two or more given points. The position vectors of the points are used to
determine the displacement vector.
Magnitude of a Vector
Let . The magnitude(length) or modulus of
Scalar or Dot Product
Let
The Dot or Scalar product of is given by
Angle between two Vectors
Let
If the angle between is then
Or
June 2016
June 2014