Vectors

1
Vectors

• Chapter 3 Pages 38-57

2
Overview

• A vector is a quantity that can be measured in
  both a magnitude and a direction

• Common Scalar Quantities
• Mass
• Work
• Energy
• Power
• Temperature
• Electric charge

• Common Vector Quantities
• Displacement
• Velocity
• Acceleration
• Force

3
Geometric Addition and Subtraction of Vectors

• Commutative Law
• a b b a
• Associative Law
• (a b) c a (b c)
• Vector Subtraction
• d a b a (-b)

4
Components of a Vector Resolving the Vectors

• ax a cos(f)
• ay a sin(f)
• a v(ax2 ay2)
• tan(f) ay/ax

5
Unit Vector Notation

• a ax i ay j az k
• b bx i by j bz k

Addition by Components
r a b rx ax bx ry ay by rz az bz
6
MULTIPLICATION!!!

• Vector by scalar
• becomes a new vector of the magnitude of
• s(scalar) a
• the direction is the original direction of vector
  a if s is and the opposite direction if s is .
• Vector by vector
• Dot product
• Cross Product

7
Dot (scalar) product

• a b abcos(f) where f is the angle between
  the directions of a and b
• a b (axbx) (ayby) (azbz)

Example Work F d (3.0N)(2.0m)ij
(10.5N)(2.0m)ii (6J)(0) (21J)(1) 21 J
If an object experiences a constant force, F
(3.0N)i (10.5N)j in the direction of (2.0m)j,
what is the work done by the force?
8
Cross (Vector) Product

• a b absin(f) where f is the angle between
  the directions of a and b
• b a -(a b) does not follow a commutative
  rule

b a
i j k
(aybz - azby)i (azbx - axbz)j (axby -
aybx)k
bx by bz
ax ay az
9
Knowledge Questions

• 1-20

10

• What are the formulas for determining components
  of a vector?

ax acos(?) ay asin(?)
11

• a b b a is the _______ law of vector
  geometric addition.

Commutative Law
12

• When a vector is multiplied by a ______, the
  direction changes based on that ________ quantity.

Scalar, scalar
13

• In the formula ab abcos(f), f is
  __________________.

Angle between the directions of a and b.
14

• True or False
• (a) a b b a
• (b) a b b a

1. False
2. True

15

• An example of a scalar quantity is _____ and an
  example of a vector quantity is _____.
• A) mass, power C) energy, work
• B) acceleration, work D) mass, acceleration

D) mass, acceleration
16

• Give the three letters used in expressing
  direction in unit vector notation.

I hat, j hat, k hat
17

• Give the formula for vector subtraction.

d a b a (-b)
18

• A ______ quantity is expressed in both magnitude
  and direction.

Vector
19

• The process of finding the components of a vector
  is called _______ ___ ______.

Resolving the vector
20

• Values of ax and ay are given. Find the
  magnitude of a.

a (ax2 ay2)½
21

• dx -25.0 m
• dy 40.0 m
• What is the magnitude of d?

d v(dx2 dy2) v(2225 m2) 47.2 m
22

• Torque is given by the formula t r F.
• If a radius is given by (3.0 m)i (4.0 m)j and
  the F (10N)i (21N)j, find the torque.

T r F (3.0m)(21N) (4.0m)(10N)k (23Nm)k
23

• Angular momentum is expressed as l m(r v).
  If an object of mass 200g, and radius (2.0m)i
  (2.0m)j is traveling at v (2.0m/s)i (3.0m/s)j
  (4.0m/s)k, what is its momentum?

l .200kg(30 42)i (-42 20)j (-22
32)k -1.6 i 1.6 j 2.0 k
24
Free Response

• 1-10

25

• Name the two laws of vector addition and give
  their formulas.

Associative Law (a b) c a (b
c) Commutative Law a b b a
26

• Describe two differences between cross and dot
  products.

1. Dot products follow a commutative rule and cross
   products dont
2. Cross products containing vectors with no k
   direction might result in a vector in the k
   direction where dot products will not.

27

• Fg 208 N
• ? 30
• Break the force into components.

• Fg

?
Fx Fgcos(?) 180.1 N Fy Fgsin(?) 104 N
28

• A person walks 3.1 km north, 2.4 km west, and 5.2
  km south consecutively. A) what is the magnitude
  of their displacement? and b) express the
  displacement vector in unit-vector notation using
  an appropriate coordinate system.

1. v(2.4)2 (2.1)2 3.18 km
2. (3.1km)i (2.4km)j (5.2km)i (-2.1km)i
   (2.4km)j