Using Vectors
Two Displacements

A hiker walks east from camp for 2.0 km, then
northeast for 3.0 km. What is the final displacement
of the hiker?

Each individual displacement is a vector that can be
represented by an arrow.
3.0 km
2.0 km
Graphical Addition

The two vectors can be added
graphically.
  
A B  C

The tail of the second vector is
placed at the tip of the first.
4.6 km

The length and directions are
kept the same.

The result is the total
displacement. It can be
measured directly.
3.0 km
2.0 km
Commutative Property

Vectors can be shifted as long as they don’t change
direction and magnitude.

Vectors can be added in reverse order and get the
same result.
  
A B  C
  
B AC
Parallelogram


If two vectors are added from a common origin one
can be shifted to make a parallelogram.
This is the same as putting the tail to the tip.
  
A B  C
Parallel and Antiparallel

Vectors that point in the
same direction are parallel.

Vectors that point in opposite
directions are antiparallel.
Cancellation

What happens if we add two antiparallel vectors of
equal magnitude?

A

A



A  ( A)  0
The vector sum is a zero length vector. The vectors
cancel out.
Vector Subtraction

To subtract one vector, add the antiparallel vector
instead.
  
A B  D
Vector Triangles

Three vectors have the
same magnitude, L, and
form an equilateral triangle.

A

B

C

 
Find A  B
• They are already tip to tail,

so  
A  B  C

Find
 
A B
• Redraw the picture
60o
Lsin60
• The direction is up
• The magnitude is
L sin 60  L sin 60  2L sin 60
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