Calculating Magnitude and Direction of Vectors

The magnitude of vector vec is the distance between the initial point A and B and the end point Q. In symbols the magnitude of vec is written as vec3

When vectors starting coordinate and ending coordinates are given then by using distance formula, magnitude of the vector can be easily find out as :

vec2

Magnitude of a Vector

vec2

Direction of a Vector

The direction of a vector is the measure of the angle it makes with x-axis. One of the formula can be used to find the direction of a vector.
vec3
Where :
where x is the horizontal change and y is the vertical change
OR
vec2
Where where ( x1 , y1 ) is the initial point and ( x2 , y2 ) is the terminal point.

So,
vec

vec

vec

vec33

vec

Representing a vector

There are a number of ways of representing vector quantities. These include:

  1. Using bold print
  2. AB where an arrow above two capital letters denotes the sense of direction, where A is the starting point and B the end point of the vector
  3. AB or a i.e. a line over the top of letters
  4. a i.e. an underlined letter

1

The force of 9N at 45◦ shown in above figure may be represented as:
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2
The magnitude of the force is 0a Similarly, the velocity of 20m/s at −60◦ shown in above figure may be represented as:

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The magnitude of the velocity is 0b

Drawing a Vector

A vector quantity can be represented graphically by a line, drawn so that:

  1. the length of the line denotes the magnitude of the quantity, and
  2. the direction of the line denotes the direction in which the vector quantity acts.


An arrow is used to denote the sense, or direction, of the vector. The arrow end of a vector is called the ‘nose’ and the other end the ‘tail’.
For example, a force of 9Nacting at 45o to the horizontal is shown in Following figure.
1
Note that an angle of+45o is drawn from the horizontal and moves anticlockwise. A velocity of 20m/s at −60o is shown in Following figure. Note that an angle of −60o is drawn from the horizontal and moves clockwise.
2

Scalars and Vectors

Any object that is acted upon by an external force will respond to that force by moving in the line of the force. However, if two or more forces act simultaneously, the result is more difficult to predict; the ability to add two or more vectors then becomes important.
 

The time taken to fill a water tank may be measured as, say, 50s. Similarly, the temperature in a room may be measured as, say, 16oC, or the mass of a bearing may be measured as, say, 3 kg.
 

Quantities such as time, temperature and mass are entirely defined by a numerical value and are called scalars or scalar quantities.
 

Not all quantities are like this. Some are defined by more than just size; some also have direction. For example, the velocity of a car is 90 km/h due west, or a force of 20N acts vertically downwards, or an acceleration of 10m/s2 acts at 50o to the horizontal.
 

Quantities such as velocity, force and acceleration, which have both a magnitude and a direction, are called vectors.

Scalar Quantities Vector Quantities
Length, Area, Volume Displacement, direction
Speed Velocity
Mass, Density Acceleration
pressure Momentrum
Temperature Force
Energy Lift, Drag
Entropy Thrust
Work, Power Weight
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