Trigonometric Method
of Adding Vectors
Analytic Method of Addition
Resolution of vectors into components:
YOU MUST
KNOW & UNDERSTAND
TRIGONOMETERY TO
UNDERSTAND THIS!!!!
Vector Components
• Any vector can be expressed as the sum of
two other vectors, called its components.
Usually, the other vectors are chosen so that
they are perpendicular to each other.
• Consider the vector V in a plane (say, the xy plane)
• We can express V in terms of
COMPONENTS Vx , Vy
• Finding THE COMPONENTS Vx & Vy
is EQUIVALENT to finding 2 mutually
perpendicular vectors which, when added
(with vector addition) will give V.
• We can express any vector V in terms of
COMPONENTS Vx , Vy
• Finding Vx & Vy is EQUIVALENT to
finding 2 mutually perpendicular vectors
which, when added (with vector addition)
will give V.
• That is, we want to find Vx & Vy such that
V  Vx + Vy (Vx || x axis, Vy || y axis)
Finding Components 
“Resolving into Components”
•Mathematically, a
component is a projection
of a vector along an axis.
– Any vector can be
completely described
by its components
• It is useful to use
Rectangular Components
–These are the projections of the
vector along the x- and y-axes
V is Resolved Into Components: Vx & Vy
V  Vx + Vy (Vx || x axis, Vy || y axis)
By the parallelogram method, clearly
THE VECTOR SUM IS: V = Vx + Vy
In 3 dimensions, we also need a Vz.
Brief Trig Review
• Adding vectors in 2 & 3 dimensions using
components requires TRIG FUNCTIONS
• HOPEFULLY, A REVIEW!!
– See also Appendix A!!
• Given any angle θ, we can construct a right triangle:
h
o
a
Hypotenuse  h, Adjacent side  a
Opposite side  o
• Define the trig functions in terms of h, a, o:
= (opposite side)/(hypotenuse)
= (adjacent side)/(hypotenuse)
= (opposite side)/(adjacent side)
[Pythagorean theorem]
Trig Summary
• Pythagorean Theorem: r2 = x2 + y2
• Trig Functions: sin θ = (y/r), cos θ = (x/r)
tan θ = (y/x)
• Trig Identities: sin² θ + cos² θ = 1
• Other identities are in Appendix B & the back cover.
Signs of the Sine, Cosine & Tangent
Trig Identity: tan(θ) = sin(θ)/cos(θ)
Inverse Functions and Angles
• To find an angle, use
an inverse trig
function.
• If sin = y/r then
 = sin-1 (y/r)
• Also, angles in the triangle add up to 90°
 +  = 90°
• Complementary angles
sin α = cos β
Using Trig Functions to Find
Vector Components
We can use all of this to
Add Vectors
Analytically!
Pythagorean
Theorem
Components of Vectors
• The x- and y-components
of a vector are its
projections along the xand y-axes
• Calculation of the x- and
y-components involves
trigonometry
Ax = A cos θ
Ay = A sin θ
Vectors from Components
• If we know the components,
we can find the vector.
• Use the Pythagorean
Theorem for the magnitude:
• Use the tan-1 function to
find the direction:
Example
V = Displacement = 500 m, 30º N of E
Example
• Consider 2 vectors, V1 & V2. We want V = V1 + V2
• Note: The components of each vector are one-
dimensional vectors, so they can be added arithmetically.
We want the sum V = V1 + V2
“Recipe” for adding 2 vectors using trig & components:
1. Sketch a diagram to roughly add the vectors graphically.
Choose x & y axes.
2. Resolve each vector into x & y components using sines &
cosines. That is, find V1x, V1y, V2x, V2y. (V1x = V1cos θ1, etc.)
3. Add the components in each direction. (Vx = V1x + V2x, etc.)
4. Find the length & direction of V by using:
Adding Vectors Using Components
•We want to add two
vectors:
•To add the vectors, add
their components
Cx = Ax + Bx
Cy = Ay + By
• Knowing Cx & Cy, the magnitude and
direction of C can be determined
Example
A rural mail carrier leaves the post office & drives
22.0 km in a northerly direction. She then drives
in a direction 60.0° south of east for 47.0 km.
What is her displacement from the post office?
Solution, page 1
A rural mail carrier
leaves the post office
& drives 22.0 km in
a northerly direction.
She then drives in a direction 60.0° south of east for 47.0 km. What is her
displacement from the post office?
Solution, page 2
A rural mail carrier
leaves the post office
& drives 22.0 km in
a northerly direction.
She then drives in a direction 60.0° south of east for 47.0 km. What is her
displacement from the post office?
Example
A plane trip involves 3 legs, with 2 stopovers: 1) Due east
for 620 km, 2) Southeast (45°) for 440 km, 3) 53° south of
west, for 550 km. Calculate the plane’s total displacement.
Solution, Page 1
A plane trip involves 3 legs, with 2 stopovers: 1) Due east for 620 km, 2) Southeast
(45°) for 440 km, 3) 53° south of west, for 550 km. Calculate the plane’s total
displacement.
Solution, Page 2
A plane trip involves 3 legs, with 2 stopovers: 1) Due east for 620 km, 2) Southeast
(45°) for 440 km, 3) 53° south of west, for 550 km. Calculate the plane’s total
displacement.
Problem Solving
You cannot solve a vector problem
without drawing a diagram!
Another Analytic Method
• Uses Law of Sines & Law of Cosines from trig.
• Consider an arbitrary triangle:
c
α
β
b
a
γ
• Law of Cosines: c2 = a2 + b2 - 2 a b cos(γ)
• Law of Sines: sin(α)/a = sin(β)/b = sin(γ)/c
• Add 2 vectors: C = A + B. Given A, B, γ
C
α
β
B
B
A
A
γ
• Law of Cosines: C2 = A2 + B2 -2 A B cos(γ)
Gives length of resultant C.
• Law of Sines:
sin(α)/A = sin(γ)/C, or sin(α) = A sin(γ)/C
Gives angle α