Transcript Vectors - Cloudfront.net

Vectors
Addition
&
Subtraction
Graphical
&
Analytical
What is a Vector?
• A quantity that requires magnitude and
direction to completely describe it is a vector.
• An arrow is used to represent a vector. The
length of the arrow represents the magnitude.
• Some examples of vectors are:
displacement, velocity,acceleration, force,
momentum
• Some examples of non-vectors (scalars) are:
distance, speed, work, energy
Vector Addition
• When two or more vectors are added the
directions must be considered.
• Vectors may be added Graphically or
Analytically.
• Graphical Addition requires the use of rulers
and protractors to make scale drawings of
vectors tip-to-tail. (Less accurate)
• Analytical Addition is a strictly mathematical
method using trigonometric functions (sin,
cos, tan) to add the vectors together.
Vector addition- Graphical
Given
vectors
and B
Re-draw as tip to
tail by moving one
of the vectors to the
tip of the other.
A
B
R
B
A
A
R = Resultant vector which is the vector
sum of A+B.The resultant always goes
from the beginning (tail of first vector) to
the end (tip of last vector).
Graphical Addition of
Several VectorsNote:  represents the angle
between vector B and the
horizontal axis
B
Given
vectors
A, B,
and C

C
A
Re-draw vectors tip-to-tail
C
B
R

A
R =A+ B +C
 (direction of R)
Vector Subtraction
• To subtract vectors, add a negative
vector.
• A negative vector has the same
magnitude and the opposite direction.
Example:
A
-A
Note: A + (-A) = 0
Vector Subtraction
Given :
Vectors A
and B, find
R=A- B
B

A
Re-draw tip to tail as A+(-B)
A

R
-B
Note: In addition,
vectors can be added in
any order. In
subtraction, you must
consider the order so
that the right vector is
reversed when added.
Equilibrant
• The equilibrant vector is the vector that
will balance the combination of vectors
given.
• It is always equal in magnitude and
opposite in direction to the resultant
vector.
Equilibrant
Given
vectors A
and B,
find the
equilibrant
Re-draw as tip to tail, find
A
resultant, then draw equilibrant
equal and opposite.
B
R
A
E
B
Right Triangle Trigonometry
hypotenuse
C
opposite
B

A
sin  = opposite
hypotenuse
cos  = adjacent
hypotenuse
tan  = opposite
adjacent
adjacent
And don’t forget:
A2 + B2 = C2
Vector Resolution
y
Given:
vector
A at angle 
from
horizontal.
Resolve
A into its
components.
(Ax and Ay)
A

Ay
x
Ax
Evaluate the triangle using sin and cos.
cos =Ax/A so… Ax = A cos
sin= Ay/A so… Ay = A sin
Hint: Be sure your calculator is in degrees!
Vector Addition-Analytical
To add vectors mathematically:
– resolve the vectors to be added into their x- and
y- components.
– Add the x- components together to get a resultant
vector in the x direction
– Add the y- components together to get a resultant
vector in the y direction
– Use the pythagorean theorem to add the resultant
vectors in the x- and y-components together.
– Use the tan function of your resultant triangle to
find the direction of the resultant.
Vector Addition-Analytical
Example:
Given: Vector A is 90 at 30O and vector B is 50 at 125O.
Find the resultant R = A + B mathematically.
B
By
Ax= 90 cos 30O = 77.9
A
55O
Bx
Ay
30O
Ax
First, calculate the
x and y components
of each vector.
Ay= 90 sin 30O = 45
Bx = 50 cos 55O =28.7
By = 50 sin55O = 41
Note: Bx will be negative
because it is acting along
the -x axis.
Vector Addition-Analytical
(continued)
Find Rx and Ry:
Rx = Ax + Bx
Ry = Ay + By
Rx = 77.9 - 28.7 = 49.2
R
Ry

Rx
Ry = 41 + 45 = 86
Then find R: R2 = Rx2 + Ry2
R2 = (49.2)2+(86)2 , so… R = 99.1
To find direction of R:  = tan-1 ( Ry / Rx )
 = tan-1 ( 86 / 49.2 ) = 60.2O
Stating the final answer
*
ccw =
counterclockwise
cw =
clockwise
• All vectors must be stated with a
magnitude and direction.
• Angles must be specified according
to compass directions( i.e. N of E) or
adjusted to be measured from the
+x-axis(0°).
• The calculator will always give the
angle measured from the closest
horizontal axis.
• CCW angles are +, CW are - *