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Vectors

Charchit Jain
23/06/2017 0 0

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.

                                   Magnitude of 3D Vector

  • A vector can also be 3-dimensional.

  • The following video gives the formula, and some examples of finding the magnitude, or length, of a 3-dimensional 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. 


                                           equal vectors

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.

                                         negative vectors

Example:

                                                   negative vectors

Vector Addition:

  • "Nose-to-Tail" Method

  • Vectors can be added using the ‘nose-to-tail’ method or "head-to-tail" 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.

Example:

Find the sum of the two given vectors a and b.

Solution:

Draw the vector a. Draw the ‘tail’ of vector b joined to the ‘nose’ of vector a. The vector a + b is from the ‘tail’ of a to the ‘nose’ of b.

Example:

Given find the sum of the vectors.

Solution:

                            

 

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