# Vectors by 7ROYwz6

VIEWS: 14 PAGES: 17

• pg 1
```									Vectors
Scalars & Vectors
• Vectors
– Quantity with both
magnitude & direction
elementary
arithmetic/algebra rules
Direction/Angle
– Examples – position,       Tail

force, moment,             Line of Action
velocities, acceleration
Parallelogram Law
• The resultant of two forces
can be obtained by
– Joining the vectors at their
tails
A
A+B
 Constructing a
B
parallelogram
 The resultant is the diagonal of the parallelogram
Triangle Construction
• The resultant of two
forces can be obtained
by
– Joining the vectors in
tip-to-tail fashion         A       B
R
   The resultant extends
from the tail of A to
• Does
A+B       =    B+A           ?

A         B
R                       R
B           A

YES! - commutative
Vector Subtraction
A-B               =   A + (-B)

A                        -B
B

-B
R
A
Vector Subtraction
• Does
A–B      =    B-A             ?

-B                     B
R
A              -R      -A

NO! – opposite sense
Vector Operations
• Multiplication & Division of Vector (A) by
Scalar (a)
a * A = aA

2A
2 * A = 2A        A

-.5 * A = -.5A     A        -.5A
Representation of a Vector
Given the points A( x1 , y1 , z1 ) and B( x2 , y2 , z2 ),
the vector a with representation AB is
a  x2  x1, y2  y1, z2  z1

Find the vector represented by the directed line segment with
initial point A(2,-3,4) and terminal point B(-2,1,1).

a  2  2,1  (3),1  4

a  4, 4, 3
Magnitude of a vector

Determine the magnitude of the following:
Example
If a  4,0,3 and b  2,1,5 , find a and the vectors a+b, a-b, 3b,and 2a+5b.

a  b  4, 0,3  2,1,5
a  4  0  3  25  5
2    2   2

a  b  4  2, 0  1,3  5
a  b  2,1,8

3b  3 2,1,5  3(2),3(1),3(5)  6,3,15
a  b  4, 0,3  2,1,5
a  b  4  (2), 0  1,3  5
2a  5b  2 4, 0,3  5 2,1,5
a  b  6, 1, 2
2a  5b  8, 0, 6  10,5, 25
2a  5b  2,5,31
Parallel
•   Two vectors are parallel to each other if one is the scalar multiple of the other.

Determine if the two vectors are parallel

These are parallel since                        These are not parallel
b= -3a                                          since 4(1/2) =2 , but
10(1/2)=5 not -9
Unit vectors
Any vector that has a magnitude of 1 is considered a unit vector.

Can you think of a unit vector?

i= 1,0,0                     j= 0,1,0                       k= 0,0,1
Standard Basis Vectors
If a= a1 , a2 , a3 , then we can write
a= a1 , a2 , a3  a1 , 0, 0  0, a2 , 0  0, 0, a3

a=a1 1,0,0  a2 0,1,0  a3 0,0,1
a=a1i  a2 j  a3k
Example- Write 1, 2,6 in terms of the standard basis vector i,j,k.

1, 2,6  i - 2j  6k
Example
If a = i + 2j - 3k and b = 4i + 7k, express the
vector 2a+3b in terms of i,j,k.

2a+3b=2(i + 2j - 3k)+3(4i + 7k)
2a+3b=2i + 4j - 6k+ 12i + 21k
2a+3b=14i+4j+15k
Unit Vectors
1    a
The unit vector in the same direction of a is   u=     a=
a    a
Find a unit vector in the same direction as 2i – j – 2k.
We are looking for a vector in the same direction as the original vector, but is also
a unit vector.
Let’s first find the magnitude   2i - j - 2k  22  (1)2  (2)2  9  3
1   a
u= a=                          Check?
a   a
Same direction?
1
u= a                           Magnitude = 1?
3
1
u= (2i - j - 2k)
3
2 1         2
u= i - j - k
3 3         3
Homework
• P649
– 4,5,7,9,11,15,17,19

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