May 87:30 AM
Bellwork
a.) Write the line 2x - 4y = 9 into slope intercept form
b.) Find the slope of the line parallel to part a
c.) Find the slope of the line perpendicular to part b or a
May 1512:20 PM
Day 1
I. Vector Vocabulary
A.) Scalar- This has magnitude but no direction.
B.) Vectors- These have both magnitude and direction.
Examples
Forces, accelertion, momentum, weight and translations in geometry
https://www.youtube.com/watch?v=fVq4_HhBK8Y
May 1810:03 AM
C.) Equality
2 or more vectors are equal if they have the same magnitude and direction.
D.) Zero Vector
E.) Addition
Let a = and b = then a + b =
F.) Subtraction 0 =
Let a = and b = then a - b =
Apr 201:30 PM
G.) Scalar Multiplication
Let a = and k is a scalar then ka =
Examples
Let a = b =
Find
a.) 5 a b.) -b + 3a
1.)
May 1810:03 AM
II.) Position vectors
Position Vector- a vector with the additional property that it is fixed at its back end to the origin.
B( b1,b2)
A( a1,a2)
AB = AO + OB = b - a
So AB =
May 1810:10 AM
1.) Given P( 2,5) and Q(3,-1)
a.) write P and Q as a position vectors
2.) Given AB = and BC =
find AC
3.) What if BA = and BC =
find AC
May 1512:21 PM
III. Magnitude
A.) Magnitude is the length or the magnitude of a vector is the absolute value.
Formula
| a | = | v| = =
**only 3 dimensional given on IB
remind you of distance formula?
May 193:05 PM
Examples
1.) Find the length of the following vectors
a.)
b.)
Apr 201:52 PM
2.) Given A= ( 3,-2) and B=( 1,2)
a.) OA
b.) AB
c.) AB try...
May 211:32 PM
Let a = 2 and b = 7 9 -3 Find
a.) | a | b.) | b| c.) | a + b|
d.) |a| + |b| e.) Show that | a + b| < |a| + |b|
Bellwork
May 1512:21 PM
A.) Magnitude is the length or the magnitude of a vector is the absolute value.
Formula
| a | = | v| =
| a | = | v| =
**only 3 dimensional given on IB Day 2
III. 3 Dimensional
yesterday
=
Apr 269:48 AM
B.) Distance and midpoint 1.) recall 2 dimensional 2.) 3 dimensional
AB =
[AB] = x2 + x1
2
y2 + y1
, 2 , z1 + z2 2
So USE FORMULA sheet
( )
May 193:05 PM
Examples
1.) Find the length of the following vectors a.)
b.)
Think about what we did yesterday
If they don't ask in words, then they use | |
Apr 201:52 PM
2.) Find the position vectors given
A= ( 3,-2,5) and B=( 1,0,-2) a.) OA
b.) AB
c.) BA
3.) Given A= ( 2,2,4) and B =( -1,1,2) find
a.) AB b.) AB
d.) [AB]
c.) [AB]
You try....
Apr 2512:21 PM
P= ( -1,2,4) and Q=( 0,2,5)
a.) What are P and Q? Points or Vectors
b.) Find PQ
c.) Find |PQ|
d.) Find [PQ]
Bellwork
Apr 47:22 AM
Slope Formula
Parallel lines
Perpendicular lines
y= 2x + 3 2x + y = 5 2x - y = 6
y = 3x - 1 3y = x + 2 3y = 4 - x
May 192:55 PM
IV. Parallel vectors
A.) Vectors a and b are said to be parallel if there is a scalar number k such that
a = kb.
Examples
1.) Show that a = -2 0 3
and b = -6 0 9
are parallel.
Day 3
May 192:57 PM
4 -1 5
a b -15
3.) Let a = 2 and b = 3
1
Find the scalar factor k where a = k b.
Apr 2612:19 PM
Bellwork
Let a =
and b =
Find x and y where a and b are parallel
May 1512:20 PM
Day 4
II. Unit Vectors
A.) A Unit vector is a vector with a length of 1.
ex:
B.) A zero vector is a vector quantity with no direction.
ex
i = 1 0 2D
j= 0 1
0 0
recall distance formula or magnitude
3D
i = 1 0 0
j = 0 1 0
k = 0 0 1
component form
May 1512:21 PM
a i + b j = ( ai + bj)
This vector will have length 1.
Examples
Find a unit vector in the same direction as
1.) 5 i - 2 j 2.) -3
6 4 1.) v = v1
v2
= v1 i + v2j unit vector form
3.) Find the value of x when x is a unit vector ¼
= 1 v |v|
= v |v|
Apr 2610:07 AM
4.) Finding a unit vector, b, in the same direction
Example
Find a unit vector, b, if the length is 5 in the same direction as
Formula k a |a|
May 1512:22 PM
1.) Find the unit vector that are parallel to
a.)
b.)
c.) 6i - 5j C.) Parallel unit vectors
± k a |a|
formula =
May 133:23 PM
Bellwork
Given the following vector find, in simplest radical form :
a.) a unit vector
b.) a unit vector traveling in the opposite direction
c.) a vector traveling in the same direction as the given but having length of 6
May 288:43 AM
Bellwork
find the length:
u = 2i - 6j +2k
May 1512:22 PM
VIII Products
A.) Inner Product, scalar , or dot product
Let v =
and w =
v w = v1 w1 + v2 w2
true also for 3 dimensions
Let v = v1
v2 v3
and w = w1
w2 w3
* given on IB exam
v w = v1 w1 + v2 w2
May 211:37 PM
Examples
Find the inner product of a and b where
1.) a = ( 3,5) and b = ( 8,-3)
2.) a = ( 2,-1,3) and b = ( 5,3,0)
May 1512:22 PM
B.) Angles
v w = |v| |w| cos θ
This is used to find angles between vectors.
rework the formula
v w = v1 w1 + v2 w2
so
v1 w1 + v2 w2
= cos θ
|v | | w|
or simply
May 211:42 PM
Find the angle between the vectors v= -i + 3j and w = -i + 2j First
Find the inner product
Second
Find the magnitude of each
Third
Plug into formula
v1 w1 + v2 w2
= cos θ
|v | | w|
May 211:44 PM
v1 w1 + v2 w2 + v3 w3 = cos θ
|v | | w|
example #2 0 -5 4
-5 -1 -3
Apr 411:55 AM
bellwork
What is the angle between the vectors
and
May 1512:23 PM
If the inner product is equal to zero, then the vectors will be perpendicular.
Why?
graph y = cos x
Examples:
1.) Determine if a and b are perpendicular vectors
a = b=
2.) Determine if a and b are perpendicular vectors
a= b=
Jun 19:57 AM
5.) Given the vertices of triangle KLM K( 4,-2,7) , L( 6,1,-1), and M ( 3,2,-2), find
a.) the position vectors KL and LM
b.)measure of < KLM
4.) Find the value of x which will make the vectors perpendicular?
May 168:57 AM
Given the vertices of ABC 6.)
using scalar products, determine if it is a right triangle.
A(-2,6) B(4,5) C(1,-4)
Apr 2512:23 PM
Find the values of x which will make the 2 vectors perpendicular
Bellwork
Apr 261:48 PM
slope = gradient
direction vector =
notice relationship with slope
Examples
1.) Given two lines 2x - y = 6 and x + 3y = 4.
Find the angle between these 2 lines.
Apr 261:51 PM
2.) same question use this time x - y= 3 and 3x + 2y = 11
Jun 19:59 AM
Bellwork
Two vectors are defined as a = 2i + xj and b = i - 4j
Find the value of x if 1.) the vectors were parallel
2.) the vectors were perpendicular
May 219:30 AM
bellwork
1.) Find p + q 2.) p - 1/2 q
3.) 4.) |p + q|
p = q =
May 153:00 PM
Bellwork
Find the exact value of:
Given:
and
May 168:57 AM
Given the vertices of ABC
using scalar products, determine if it is a right triangle.
A(-2,6) B(4,5) C(1,-4)
Bellwork
