Transcript Definitions

Vectors
An Introduction
There are two kinds of
quantities…
• Vectors are quantities that
have both magnitude and
direction (e.g., displacement,
velocity, acceleration).
• Scalars are quantities that
have magnitude only (e.g.,
position, speed, time, mass).
Notating vectors
•Vector:
R
→
R
This is how you draw a vector.
tail
R
head
Notating scalars
•Scalar:
R
There is no standard
way to draw a scalar!
Direction of Vectors
θ
B
A
x
θ
x
Vector angle ranges
II
90o < θ < 180o
y
I
0 < θ < 90o
θ θ
θ θ
III
180o < θ < 270o
x
IV
270o < θ < 360o
Magnitude of Vectors
• The best way to describe the
magnitude of a vector is to
measure the length of the vector.
• The length of the vector is
proportional to the magnitude of
the quantity it represents.
Magnitude of Vectors
A
If vector A represents
a displacement of three
miles to the north…
B
Then vector B, which is
twice as long, would
represent a displacement
of six miles to the north!
Equal Vectors
Equal vectors
have the same
length and
direction, and
represent the
same quantity
(such as force
or velocity).
Inverse Vectors
Inverse
vectors
have the
same
length, but
opposite
direction.
A
-A
Vectors: x-component
Ax = A cos θ
A
θ
Ax
x
Vectors: y-component
Ay = A sin θ
A
θ
Ay
x
Vectors: angle
y
θ = tan-1 (Ry/Rx)
Ry
θ
Rx
x
Vectors: magnitude
y
R = √ (Rx2 + Ry2)
R
Ry
Rx
x
Graphical Addition of
Vectors
You’ll need:
Graph paper
Pencils
Ruler
Protractor
Graphical Addition of Vectors
B
A
R
A+B=R
R is called the resultant vector!
The Resultant and the
Equilibrant
The sum of two or more vectors is
called the resultant vector.
The resultant vector can replace the
vectors from which it is derived.
The resultant is completely canceled
out by adding it to its inverse,
which is called the equilibrant.
Graphical Addition of Vectors
B
A
E
R
A+B=R
E is called the equilibrant vector!
Component Addition of Vectors
1) Resolve each vector into its x- and y-
components.
Ax = Acosθ
Bx = Bcosθ
Cx = Ccosθ
Ay = Asinθ
By = Bsinθ
Cy = Csinθ
etc.
2) Add the x-components (Ax, Bx, etc.)
together to get Rx and the ycomponents (Ay, By, etc.) to get Ry.
Component Addition of Vectors
3) Calculate the magnitude of the
resultant with the Pythagorean
Theorem R = √(Rx2 + Ry2)
4) Determine the angle with the
equation θ = tan-1 Ry/Rx.
Relative Motion
S = swimmer
W = water
Vs
Vt = Vs + Vw
Vw
Relative Motion
Vs
Vt = V s + Vw
Vw
Relative Motion
Vs
Vt = Vs + Vw
Vw