Adding Vectors

        We can add vectors 
                by components

is the vector sum (or resultant) of  the vector and

 Adding vectors geometrically (two-dimensional)

A procedure for adding two-dimensional vectors and geometrically.

(1) On paper, sketch vector to some convenient scale and at the proper angle.
(2) Sketch vector
to the same scale, with its tail at the head of vector a, again at the proper angle.
(3) The vector sum
is the vector that extends from the tail of to the head of .

Commutative law       

Associative law           

vector subtraction       

Two Equivalent way to represent a vector

  1. Magnitude-angle notation

  2. Components notation (unit-vector notation)

   Unit Vectors

Math Refresher


Question: If 

Find its magnitude and direction.










Question: A small airplane leaves an airport is landed 100 km away, in a direction making an angle of 30 east of north. How far east and north is the airplane from the airport?


The airplane is landed 50 km east and 87 km north of the airport.

Question: How many parameters do you need to determine a vector?


In two dimension (all vectors in one plane)

Two components on two perpendicular axes
or equivalently
its magnitude and an angle from an axis.

In three dimension (General)

you need three components on three perpendicular axes
or equivalently
its magnitude and two angles from two axes.


Question: A particle moves 3 m westward and 4 m southward and 2 m upward. What is its displacement?





Adding vectors components



rx = ax + bx + cx.
ry = ay + by + cy.
rz = az + bz + cz.

Question: Suppose


what is



rx = 4+8 = 12 m.
ry = 1 - 1 + 3 = 3 m.
rz =2 - 4 = -2 m



Question: Suppose

Find the magnitude c and b?




cx = ax + bx            →    c cos 450 = 3 m + b cos 800
cy = ay + by            →    c sin 450 =  b sin 800


c  =sqrt(2)( 3 m + c cot 800/sqrt(2))

c=5.2 m

b =3.7 m

Freedom in choosing a coordinate system

we have great freedom in choosing a coordinate system, because the relations among vectors do not depend on the location of the origin of the coordinate system or on the orientation of the axes. This is also true of the relations of physics; they are all independent of the choice of coordinate system.