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 , ^{z}^{1}^{ + z}^{2}
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**