What is Vector?

Scalar quantities are quantities that have only magnitudes such as time, area and distance.

Vector quantities are quantities that have both magnitudes and directions such velocity (speed and direction), force and acceleration.

A vector can be represented by a directed line segment, whose direction is given by the arrow and the length shows the magnitude of the vector.

The vectors can be denoted by or AB or or or a.

A is called the initial point and B is called the terminal point of .

The magnitude of a vector is the length of the corresponding segment. The magnitude of is denoted by .

A vector can be written as an ordered pair called a column vector.

Consider the line PQ in the diagram. The line represents the translation of P to Q, which is 2 right and 3 up.

This can be written as the ordered pair
Example:
Express as a column vector.
Solution:
The translation of C to D is 4 right and 3 down.
Vector Magnitude:
The length of a vector is called the magnitude or modulus of the vector.
Example:
Express each of the following vectors as a column vector and find its magnitude.
Vectors in 2D:

Adding vectors geometrically, scalar multiplication, how to find the magnitude and direction angle of a vector.

A vector with initial point at the origin and terminal point at (a, b) is written <a, b>.

Geometrically, a vector is a directed line segment, while algebraically it is an ordered pair.
Vectors in 3D
The following diagram shows how to find the magnitude of a 3D Vector.

A vector can also be 3dimensional.

The following video gives the formula, and some examples of finding the magnitude, or length, of a 3dimensional vector.
Equal vectors:

Equal vectors are vectors that have the same magnitude and the same direction.

Equal vectors may start at different positions. Note that when the vectors are equal, the directed line segments are parallel.
Equality Of Column Vectors:
If two vectors are equal then their vector columns are equal.
Example:
The column vectors p and q are defined by
Given that p = q
a) find the values of x and y
b) find the values of and
c) express y in terms of x
Solution:
Negative Vectors:
The negative sign reverses the direction of the vector.
Example:
Vector Addition:

"NosetoTail" Method

Vectors can be added using the ‘nosetotail’ method or "headtotail" method.

Two vectors a and b represented by the line segments can be added by joining the ‘tail’ of vector b to the ‘nose’ of vector a. Alternatively, the ‘tail’ of vector a can be joined to the ‘nose’ of vector b.