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Unit 1
Mechanics

Vectors
Topics Covered:

Any vector is space can be defined by its coordinates in terms of its x, y, and z position.

In a two dimensional frame we need only consider its x and y co-ordinates.  Therefore any vector can be thought of as a resultant when its horizontal component in the x direction is added to its vertical component in the y direction.

Figure 1:
A point P defined by its Cartesian co-ordinates
(x , y) with respect to the origin (0,0)

Figure2:
The vector V maps the point P in terms
of its position from the origin

   

Figure 3:
Vy is the vertical component of the
vector v 

Figure 4:
Vx  is the horizontal component of the
vector v 

   

Figure 5:
the x and y components of vector v

We can also define Vx and Vy in terms of  V and
the angle Ө using trigonometry


rearranging this equation we obtain:

The vector equation for vector v in terms of
 its components vx and vy is:
v = vx + vy

Here are some  examples of vector components:

1. What are the horizontal and vertical components of a plane moving at 300 km/h [E650N]?

Refer to Figure 5 above. 
Here q =  65 0  and v = 300 km/h

2. What are the components of a vector 8 cm long with bearings [W150S]?

Addition of Vectors using the components method

When adding two or more vectors using the components method follow these guidelines.

  1. Sketch the vectors
  2. Calculate each vector's horizontal (x) and vertical component (y)
  3. add all the horizontal components together (taking into account the sign of the vector)
  4. add all the vertical components together (taking into account the sign of the vector)
  5. Now you have a resultant horizontal vector in the x direction and a resultant vertical vector in the y direction.
  6. Use the resultant x and y vectors to calculate the total resultant vector.
  7. Calculate the angle and state the direction of the final (resultant) vector.
The animations below illustrate the fact that irrespective of the path one takes from an initial position to a final position the sum of all vectors will be equal to a resultant vector which has only two components: an x (horizontal) component and a y (vertical component).  Here the vectors a, b, and c all have their own  x and y components which can be added together to give a total horizontal component V1 and a total vertical component V2.  Thes, in turn can be added to find out the resultant vector VTot. Note that in both scenarios (figure A and figure B), the end point with respect to the origin is the same.
 

Figure A

Figure B

Example:
Find the final displacement of a man who first walks North for 5.0 km, then East for 3.0 km and finally [E600S] for 8.0 km.  The sketch is not to scale.

Solution:

Given: d1 = 5.0 km [N],  d2 = 3.0 km [E], d3 = 8.0 km [E600S]

Find:

Method:
set-up a table for each vector where you can enter the x and y components of each.

Vector Sketch X component (horizontal) Y component (vertical)
d1 = 5.0 km [N]

d1x = 0

d1y = +5
d2 = 3.0 km [E]

d2x = + 3


d2y = 0

 

d3 = 8.0 km [E600S]

 

Totals

dTx = 0 + 3 + 4
 = +7 km
(to the right)

dTy = 5 + 0+ (-)6.9
 = -1.9 km
 (down)
 
  Use Pythagoras' Theorem to find the magnitude of total displacement :


 

 
DIRECTION

 q = inv tan (dTy/dTx) = 150
 

Conclusion:
                         The total displacement of the man is : 7.3 km [E150]