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Vector and scalar
Chapter 1(b)
Scalar and vectors Definitions Addition n subtraction rules Scalar and dot product
Learning Outcomes
Define scalar and vector quantities, unit vector in Cartesian coordinates.
Explain vector addition and subtraction n their rules.
Define and use dot and cross product (multiplying vector)
Trigonometry
Trigonometry
h
hosin
h
hacos
a
o
h
htan
To find the length
Trigonometry
h
ho1sin
h
ha1cos
a
o
h
h1tan
To find the angle
Characteristics of a Scalar
Quantity Only has magnitude
Requires 2 things:
1. A value
2. Appropriate units
Ex. Mass: 5kg
Temp: 21 C
Speed: 65 m/s
Characteristics of a Vector
Quantity Has magnitude & direction
Requires 3 things:
1. A value
2. Appropriate units
3. A direction!
Ex. Acceleration: 9.8 m/s2 down
Velocity: 25 m/s West
Scalars and Vectors
By convention, the length of a vector
arrow is proportional to the magnitude
of the vector.
8 N 4 N
Arrows are used to represent vectors. The
direction of the arrow gives the direction of
the vector.
Scalar and Vector Quantities
The car moved a distance of 2 km in a
direction 30o north of east
3-2 The Components of a Vector
. ofcomponent vector theand
component vector thecalled are and
r
yx
y
x
The Components of a Vector
.AAA
AA
A
yx
that soy vectoriall together add and
axes, and the toparallel are that and vectors
larperpendicu twoare of components vector The
yxyx
The Components of a Vector
It is often easier to work with the scalar components
rather than the vector components.
. of
componentsscalar theare and
A
yx AA
1. magnitude with rsunit vecto are and yx
yxA yx AA
The Components of a Vector
Example
A displacement vector has a magnitude of 175 m and points at
an angle of 50.0 degrees relative to the x axis. Find the x and y
components of this vector.
r
ysin
m 1340.50sinm 175sin ry
r
xcos
m 1120.50cosm 175cos rx
yxr m 134m 112
Signs of vector components:
The components of a vector
EXERCISE
1)The vector A has a magnitude of 7.25 m.
Find its components for direction of
angles of
(a)=5.0o
(b)=125o
(c)= 245o
(d) = 335o
Answer
(a)Ax=7.22m, Ay=0.632m
(b) Ax=-4.16m, Ay=5.94m
(c) Ax=-3.06m, Ay=-6.57m
(d) Ax=6.57m, Ay=-3.06m
IMPORTANT FOR VECTOR
COMPONENTS
Given the components of a vector, find its
magnitude and direction:
x
y
yx
A
A
AAA
1
22
tan
cosAAx
sinAAy
Ay= 2.00 m
Ax = 6.00 m
EXAMPLE:
Length, angle, and components can
be calculated from each other using
trigonometry:
Given vector component x is 6.00m and vector
component y is 2.00m, find the magnitude and
direction of vector A.
A
2.00 m
6.00 m
222 m 00.6m 00.2 R
R
m32.6m 00.6m 00.2 22 R
2.00 m
6.00 m
00.600.2tan
4.1800.600.2tan 1
If each component of a
vector is doubled, what
happens to the angle of
that vector?
1) it doubles
2) it increases, but by less than double
3) it does not change
4) it is reduced by half
5) it decreases, but not as much as half
Question 3.4 Vector Components I
If each component of a
vector is doubled, what
happens to the angle of
that vector?
1) it doubles
2) it increases, but by less than double
3) it does not change
4) it is reduced by half
5) it decreases, but not as much as half
The magnitude of the vector clearly doubles if each of its
components is doubled. But the angle of the vector is given by tan
= 2y/2x, which is the same as tan = y/x (the original angle).
Question 3.4 Vector Components I
Vector Addition and Subtraction
Often it is necessary to add one vector to another.
Vector Addition and Subtraction
5 m 3 m
8 m
Vector Addition and Subtraction
A
B
BA
A
B
BA
Component Method of Vector Addition
Treat each vector separately:
1. To find the X component, you must:
Ax = Acos
2. To find the Y component, you must:
Ay = Asin
3. Repeat steps 1 & 2 for all vectors
Component Method (cont.)
4. Add all the X components (Rx)
5. Add all the Y components (Ry)
6. The magnitude of the Resultant Vector is
found by using Rx, Ry & the Pythagorean
Theorem:
R2 = Rx2 + Ry2
7. To find direction: Tan = Ry / Rx
Vector Addition and Subtraction
Adding vectors graphically: Place the tail of the second at the head of
the first. The sum points from the tail of the first to the head of the last.
yxA yx AA
Addition Rule for Two Vectors
BAC
1. Find the components of each vector to be added.
yxB yx BB
yxyxyxC
yyxx
yxyx
BABA
BBAA
xxx BAC
yyy BAC
Addition Rule for Two Vectors
2. Add the x- and y-components separately.
3. Find the resultant vector.
Subtracting Vectors
Subtracting Vectors: The negative of a vector is a vector of the same
magnitude pointing in the opposite direction. Here,
D= A B
Lets try
)5(
)22332(
)223()32())(
)35(
)22332(
)223()32())(
zyx
zyxzyx
zyxzyxBAii
zyx
zyxzyx
zyxzyxBAi
Component Method (cont.)
Lets try!
A = 2 m/s 30 N of E
B = 3 m/s 40 N of W
(this is easy!)
Find: Magnitude & Direction
Magnitude = 2.98 m/s
Direction = 79.02 N of W@
180-79.02 =100.98
Component Method (cont.)
Lets try!
F1 = 37N 54 N of E
F2 = 50N 18 N of W
F3 = 67 N 86 S of W
(this is not so easy!)
Find: Magnitude & Direction
Magnitude =37.3 N
Direction = 35.1 S of W @
180+35.1=215.1
3-4 Unit Vectors
Unit vector is dimensionless vectors of unit length (magnitude of 1) with a function to indicate direction.
Unit vector - indicates the x-direction
Unit vector - indicates the y-direction
Unit vector - indicates the z-direction
x
zy
EXERCISE
To find a magnitude of a vector
1) Find magnitude vector A and vector B respectively
2) Find magnitude of vector A +B
zyxB
zyxA
223
32
To find a magnitude of a vector
1) Find magnitude vector A and vector B respectively
units
units
2) Find magnitude of vector A +B
zyx
zyxzyxBA
35
22332)(
units
BA
9.535
135222
zyxB
zyxA
223
32
There are two distinct ways to multiply vectors, referred to as the dot product and the cross product.
The dot product yields a scalar (a number) as the result.
The cross product yields a vector as the result.
Vector Multiplication
Definition of the scalar, or dot, product:
Application example:
Work is the dot product of force and displacement
zByBxBzAyAxABA zyxzyx
zzyyxx BABABA
Dot Product
Exercise
yxb
yxa
52
86
1) Find a.b
2) Find angle between a and b
The vector cross product is defined as:
The direction of the cross product is defined by a
right-hand rule:
Cross Product
The cross product can also be written in determinant
form:
Cross Product
Application example: The relation of the magnetic force
on a charge q with a velocity in a magnetic field is v
B
sinqvBBvqF
x zy
zBABAyBABAxBABA xyyxxzzxyzzy
Exercise
zyxb
zyxa
843
562
1) Find axb
2) Find the angle between a and b