Adding Vectors
We can add
vectors
geometrically
or
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
Magnitude-angle notation
Components notation (unit-vector notation)
Question: If
Find its magnitude and direction.
Answer:
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?
Answer:
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?
Answer:
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?
Answer:
Adding vectors components
Suppose
Then
r_{x }= a_{x }+ b_{x} + c_{x}.
r_{y }= a_{y }+ b_{y} + c_{y}.
r_{z }= a_{z }+ b_{z} + c_{z}.
Question: Suppose
what is
Answer:
r_{x }= 4+8 = 12 m.
r_{y }= 1 - 1 + 3 = 3 m.
r_{z }=2 - 4 = -2 mOr
Question: Suppose
Find the magnitude c and b?
Answer:
c_{x }= a_{x }+ b_{x }→ c cos 45^{0} = 3 m + b cos 80^{0 }c_{y }= a_{y }+ b_{y }→ c sin 45^{0} = b sin 80^{0}
c =sqrt(2)( 3 m + c cot 80^{0}/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.