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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).
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?
(Here I have omitted the arrows on top of
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]
Solution: dt = 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 VT 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 VT 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:
A pigeon flies 5 km [N300E], then 2.5 km [E400S]. Calculate its final displacement.
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