Download Components of a vector

Vector Operations
Chapter 3 section 2
A+B=?
B
A
Vector Dimensions
-
When diagramming the motion of an
object, with vectors, the direction and
magnitude is described in x- and ycoordinates simultaneously.
-
This allows vectors to be used for 1-d and 2-d
motion.
How can I get to the red dot starting from
the origin and can only travel in a straight
line?
y
x
There are 3 main
different ways that I can
travel to get from the
origin to the red dot by
only traveling in a
straight lines.
y
x
Solving For The Resultant of 2
Perpendicular Vectors
When two vectors are perpendicular to
each other it forms a right triangle, when
the resultant is formed.
 Right triangles have special properties that
can be used to solve specific parts of the
triangle.

 Such
as the length of sides and angles.
Magnitude of a Vector

To determine the magnitude of two vectors,
the Pythagorean Theorem can be used
 As
long as the vectors are perpendicular to each
other.
Pythagorean Theorem
c²=a²+b²
(length of hypotenuse)²=(length of leg)²+(length of other leg)²
Applied Pythagorean Theorem
Δx
c
a
Δy
R
b
c2=a2+b2
(Mathematics)
R²=Δy²+Δx²
(Physics)
Direction of a Vector

To determine the direction of the vector,
use the tangent function.
Tangent Function
Tanθ=opp/adj
opp
θ
adj
Applied Tangent Function
Δx
c
a=opp
θ
Δy
R
θ
b=adj
𝑜𝑝𝑝
𝑇𝑎𝑛 𝜃 =
𝑎𝑑𝑗
(Mathematics)
Δ𝑦
𝑇𝑎𝑛 𝜃 =
Δ𝑥
(Physics)
Recall Vector Properties
Δx
Δy
R
θ
=
R
θ
Δx
Δy
Example Problem

A soldier travels due east for 350 meters
then turns due north and travels for
another 100 meters. What is the soldiers
total displacement?
Example Picture
Example Work
Example Answer

R=364 m @ 15.95°
Vector Components

Every vector can be broken down into its
x and y components regardless of its
magnitude or direction.
Vectors Pointing Along a Single Axis

When a vector points along a single axis,
the second component of motion is equal
to zero.
Vectors That Are Not Vertical or
Horizontal

Ask yourself these questions.
 How
much of the vector projects onto the x-
axis?
 How much of the vector projects onto the yaxis?
Components of a Vector
y
A
A
θ
A
x
x
x
Resolving Vectors into Components

Components of a vector – The projection
of a vector along the axis of a coordinate
system.
 x-component
is parallel to the x-axis
 y-component is parallel to the y-axis
 These components can either be positive or
negative magnitudes.

Any vector can be completely described by
a set of perpendicular components.
Vector Component Equations

Solving for the x-component of a vector.
 𝐴𝑥


= 𝐴𝑐𝑜𝑠𝜃
(𝑥 − 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴 = 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴 •
cos(𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴)
Solving for the y-component of a vector.
 𝐴𝑦

= 𝐴𝑠𝑖𝑛𝜃
(𝑥 − 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴 = 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴 •
cos(𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴)
Example Problem

Break the following vector into its x- and
y- components.
A = 6.0 m/s @ 39°
Example Problem Work
A = 6.0 m/s @ 39°
Example Problem Answer
Ax = 4.66 m/s
 Ay = 3.78 m/s

Example Problem:

A plane takes off from the ground at an
angle of 15 degrees from the horizontal
with a velocity of 150mi/hr. What is the
horizontal and vertical velocity of the
plane?
Example Picture
Example Work
Example Answer

Horizontal velocity = 144.89 miles per hour
 Vx=144.89mi/hr

Vertical velocity = 38.82 miles per hour
 Vy=38.82mi/hr
Adding Non-Perpendicular Vectors

When vectors are not perpendicular, the
tangent function and Pythagorean
Theorem can’t be used to find the
resultant.
 Pythagorean
Theorem and Tangent only work
for two vectors that are at 90 degrees (right
angles)
Non-Perpendicular Vectors

To determine the magnitude and direction of the
resultant of two or more non-perpendicular
vectors:


Break each of the vectors into it’s x- and ycomponents.
It is best to setup a table to nicely organize your
components for each vector.
Component Table
x-component
Vector A - (A)
Vector B - (B)
Vector C - (C)
Add more rows if
needed
Resultant - (R)
y-component
Non-Perpendicular Vectors

Once each vector is broken into its x- and ycomponents :

The components along each axis can be added
together to find the resultant vector’s components.
 Rx
= Ax + Bx + Cx + …
 Ry = Ay + By + Cy + …
 Only then can the Pythagorean Theorem and
Tangent function can be used to find the
Resultant’s magnitude and direction.
Example Problem

During a rodeo, a clown runs 8.0m north,
turns 35 degrees east of north, and runs
3.5m. Then after waiting for the bull to
come near, the clown turns due east and
runs 5.0m to exit the arena. What is the
clown’s total displacement?
Practice Problem Picture

Step #1: Draw a picture of the problem
Practice problem Work

Step #2: Break each vector into its x- and
y- components.
x-component
Vector A - (A)
Vector B - (B)
Vector C - (C)
Resultant - (R)
y-component
Step #3: Find the resultant’s components by
adding the components along the x- and yaxis.
x-component
Vector A - (A)
Vector B - (B)
+
Vector C - (C)
Resultant - (R)
y-component
Step #4: Find the magnitude of the vector
by using the Pythagorean theorem.
R2 = Δx2 + Δy2
Step #5: Find the direction of the vector by
using the tangent function.
Tan θ = Δy/Δx
Step #5: Complete the final answer for the
resultant with its magnitude and direction.

Practice Problem Answer
 Resultant
displacement = 12.92m @ 57.21º