Chapter 3 Vectors in Physics
1) The tail of the second vector is placed at the
arrow end of the first vector.
Adding Graphically:
tor
vec
t
n
ulta
Res
First vector
nd
o
c
Se
r
to
c
ve
3) When adding more than two vectors, always
place tail of next vector to arrow of previous
vector and draw resultant vector last.
Parallelogram Technique:
(1)
r
ecto
v
t
n
ulta
Res
(2)
First vector
d
on
c
Se
2) The resultant vector starts at the tail of the
first vector and finishes on the arrow end of the
last vector.
r
to
c
ve
1) Draw a line [labeled (1) on the figure] that is
parallel to the first vector and that intersects the
second vector at its arrow end .
2) Draw a line [labeled (2) on the figure] that is
parallel to the second vector and that intersects
the first vector at its tail .
3) The resultant will be the diagonal across this
parallelogram - starting at the tail of the first
vector and finishing on the arrow end of the
second vector.
The two graphical techniques for adding vectors work for any number of vectors (the
parallelogram technique is applied to two vectors at a time), but the accuracy of the
resultant vector depends strongly on the care taken in doing the drawings. The more
vectors added, the more difficult this becomes. The method of components avoids this
problem by determining the resultant vector algebraically, but both component and
graphical methods should be used so that one can provide a check on the other.
Displacement vector: The displacement vector (∆r)
depends on the difference between the final and initial
position vector of an object. To determine a position vector
you must have a coordinate system with an origin (0,0).
y
rf
In the figure at the right, ri is the initial position vector
and rf , is the final position vector. The vector labeled ∆r is
the displacement vector.
∆r
ri
(0,0)
x
Note that displacement shows how a vector subtraction (∆r = rf – ri ) can be rewritten
as an addition (since ri + ∆r = rf ). Remember that the negative of a vector is a vector
with the same length, but points in the opposite direction to the original.
Section 3.4
The use of unit vectors allow us to write an expression for a vector (or sum of vectors)
→
→
in terms of vector components. E.g., A + B = (Ax + Bx) x^ + (Ay + By) y^ .
Right triangle trigonometry:
Pythagorean theorem applied to the triangle below: c2 = a2 + b2 .
Terminology
β
c
c = hypotenuse
a = side adjacent θ
b
= side opposite β
b = side opposite θ
= side adjacent β
θ
a
Extreme values:
θ
0°
90°
Basic Trigonometry
functions:
opp.
b
sin θ = hyp. = c = cos β
adj.
a
cos θ = hyp. = c = sin β
opp.
b
tan θ = adj. = a
sin θ
0
1
cos θ
1
0
tan θ
0
+ ∞ (– ∞ for –90°)
β
Ax = A cos θ
= A sin β
Components of Vectors:
A = the magnitude of
the vector (the
hypotenuse)
A
Ax = the x component
of A
θ
Ay = the y component
of A
Ax
Ay
Ay = A sin θ
= A cos β
2
A 2
2
= Ax + Ay
Signs of components:
x
y
x
– +
y
+ +
As shown in the figure at left,
given the vector's direction you
can determine the signs (+ or –)
of the vector's components.
For example, a vector pointing toward
the second quadrant would have a
negative (–) x component and a
positive (+) y component.
x
y
– –
Section 3.6
x
y
+ –
Remember, when finding a resultant's
components, you must include the
signs of the individual vector's
components.
The relative velocity formula is written so that the order of the subscripts reminds you
to order the vector sum correctly: v1 3 = v1 2 + v2 3. Notice on the right hand side of the
equation that 2 is the subscript in common that each right hand velocity refers to.