Unit 1

Vectors

A vector is a directed line segment - a line with an arrow head at the end indicating a direction

When writing a vector we place an arrow on top of the symbol to distinguish it from its scalar counterpart.

The symbol for displacement is
Displacement is  a vector quantity.  
The standard SI (System International) unit of displacement is the meter (m). 
The symbol for distance is d (no arrow means scalar quantity).

Adding Vectors

A. Collinear Vectors:

These are vectors that are found along the same line of action: East - West or North - South.

When adding collinear vectors one must ensure that they are all pointing in the same direction by reversing the direction of one or more and changing the sign in front of the magnitude of the vector.

Example: Mr. Forget drives 5.0 km East from home to a gas station and then back West to the local bakery for another 2.0 km.  What is Mr.. Forgets' total displacement?

Solution:

Given:

(Here I have omitted the arrows on top of d₁ and d₂ for ease of editing but keep in mind that they are vectors. I will use "bold" characters instead to represent vectors.)

d₁ = 5.0 km [E]
d₂ = 2.0 km [E] = -2.0 km [W]

Find: d_t    

Solution:   d_t  = d₁ + d₂ = 5.0 km [E] + (-2.0 km [W] ) = 3.0  km [E]

Please note that the total distance is still 7.0 km.

B. Non-Collinear Vectors

When vectors are in the same plane (but not along the same line of action) they can be added using three different methods:

The general rule for adding vectors regardless of the method is: "add vectors from tail to head". 
 This means that the resultant V_T  in the following illustration is the sum of the vectors V₁ and V₂ and is drawn from the tail of the first vector (V₁) to the head of the last vector (V₂).

Standard Vector Direction

Note that V_T is in a direction other that the standard North-South-East-West directions. To report this type of vector direction we make use of the nautical directional system (the compass).

 The diagram below demonstrates how to report a vector direction properly using standard format.  The vector has direction 60⁰ East of North (or 30⁰ North of East) and a magnitude of 10 m.  In other words, to draw this vector using a protractor and a ruler, we started measuring 60⁰ from the North position and we moved towards the East position. 

In standard notation we report vector d as: =  10 m [N60⁰E].

Note: we can also report this vector as = 10 m [E30⁰N].

Method 1. Graphical Technique.

This technique requires very good accuracy in drawing vectors and one must  use proper scaling to get good results..

Step by step approach:

1. Choose a scale according to the given information
2. Start with a set of reference axis and draw the first vector using a ruler to draw its magnitude to sale and a protractor to report its direction properly as illustrated above.
3. At the head of the vector draw a small cross.  This is a new reference axis to guide you in drawing successive vectors.
4. Draw the second vector (and the third if necessary) using steps 1 to 3.
5. Draw the resultant vector from the tail of the first vector to the head of the last vector.
6. Use a ruler and protractor to report the magnitude and direction of the resultant vector properly.

Example:

A pigeon flies 5 km [N30⁰E], then 2.5 km [E40⁰S]. Calculate its final displacement.

Solution:

Step 1 - set a scale -- scale = 1cm = 1 km

therefore: a = 5 cm  [N30⁰E], and b =  2.5 cm [E40⁰S].

Step 2 - draw vectors a and b to scale

Step 3 - join the tail of vector a to the head of vector  b to find the resultant vector c.

Step 4 - measure resultant vector b using a ruler and convert to real-time units using your scale.

Step 5 - use a protractor to measure and report the direction of resultant vector c.

Step 6 - Conclusion - The final displacement  of the pigeon is c = 5.2 km [E30⁰N]

Figure 1 - Steps 1 to 4	Figure 2 - Steps 5 & 6

Method 2. Mathematical

A. Sine-Cosine Law

B. Components