Download Vectors

Vectors-1
Vectors
A vector is a mathematical object consisting of a magnitude (size) and a direction.
A vector can be represented graphically by an arrow:
direction of arrow =
direction of vector
length of arrow = magnitude of vector
G
Vector quantity written in bold (A) or with a little arrow overhead ( A )
G
A (no arrow) = A = magnitude of the vector = positive number
Examples of vector quantities: position, velocity, acceleration, force, electric field.
Same direction, same magnitude ⇒
same vector
y
A
Vector = magnitude + direction (not location)
A
A
x
y
G
In 2D, need 2 numbers to specify a vector A : A and angle θ
A
θ
x
Addition of Vectors
y
B
B
C
A
A
x
G G
G
A+B = C
Vectors-2
G G
G G
Vector addition is commutative: A + B = B + A
Graphical addition: "tip-to-tail" or "tail-to-head" method:
A
B
C
Addition by "parallelogram method" (same result as tip-to-tail method)
C
B
Can add lots of vectors (like steps in a treasure map: "take 20 steps east,
then 15 steps northwest, then…")
G G G G G
G
A+B + C+D+E = S
D
E
C
B
S
A
Negative of vector (same size, opposite direction):
A
−A
Multiplication of a vector by a number:
b =3
c = −2
bA
3 times as
long as A
A
B
Components of a Vector
G
G
G
A = Ax + Ay
y
Ay
Ax = A cos θ = x-component
Ay = A sin θ = y-component
A
θ
Ax
x
C
B
A
cB
negative c
flips direction
A
Vectors-3
Ax
⇒ A x = A cos θ
A
A
sin θ = y ⇒ A y = A sin θ
A
cos θ =
A
Ay
θ
Ax
Components of a vector can be positive or negative.
G
( A = | A | is positive always, but Ax and Ay can be + or − )
G
Here, Bx is negative, because Bx is along the −x direction.
G
By is positive, because By is along the +y direction.
y
B
By
Bx
x
Can specify a vector (in 2D) in two ways: by giving A, θ OR Ax, Ay
A =
A
Ay
θ
Ax 2 + A y2
Ay
tan θ =
Ax
Ax
Example of vector math: Ax = +2, Ay = −3 What is magnitude A, and angle with x-axis α ?
y
x
α
A
2
3
A =
tan α =
22 + 32 =
3
2
4+9 =
13 3.6
⎛ 3⎞
⇒ α = tan −1 ⎜⎜ ⎟⎟⎟ = 56.3o
⎜⎝ 2 ⎠
Vectors-4
Vector Addition by Components:
Proof by diagram:
G
G G
C = A+B ⇒
y
C x = A x + Bx
C y = A y + By
C
B
A
Bx
Ax
Cx
Vector subtraction:
G G
G
G
A − B = A + (−B)
← "substract" means "add negative of"
G G
G
Graphically: D = A − B
−B
D
−B
B
A
G G
G
D = A−B
Subtraction by components:
G G
G
D = A−B
B
A
⇒ D x = A x − Bx
D y = A y − By
D
A
G
G G
D +B = A
is same as
G G
G
D = A−B
x