Today’s Plan

• C1 Marks – Class Average: 76%

• New Lesson– Midpoint (6.4)– Division Point (6.4)– Scalar Product (6.5)

• Homework Questions

• PRACTICE

Midpoint of a Line Segment

• The coordinate of M are:

M

B

A

Properties ofMidpoint of a Line Segment

1. MA + MB = 0

2. AM = MB

3. AM = ½AB

4. AB = 2AM

M

B

A

Division point of a Line Segment

• Point P divides the line:– in a 3:2 ratio or 3/2 from A

or– in a 2:3 ration or 2/3 from B

P

B

A

2

3

Division point of a Line Segment

We can also say:P is located at 3/5 from point A

or

P is located at 2/5 from point B

P

B

A

2

3

Division point of a Line Segment

The vector equality is:AP = 3/5 AB

or

BP = 2/5 BA

P

B

A

2

3

Example 1Consider A(1,4) and B(7,1), find P if AP = ⅔ AB

AB = (7-1,1-4) = (6,-3)

AP = (xP-xA, yP-yA)

We know that AP = ⅔ABso fill in the other info…

(xP - 1, yP - 4) = ⅔ (6,-3)

Example 1Consider A(1,4) and B(7,1), find P if AP = ⅔ AB

(xP - 1, yP - 4) = ⅔ (6,-3)

(xP - 1, yP - 4) = (4,-2)

Look at x and y individually…

xP – 1 = 4 and yP – 4 = -2

xP = 4 +1 and yP = -2 +4

xP = 5 and yP = 2

So point P is (5,2)

SCALAR PRODUCTThe scalar product of two vectors u and v is

a real number defined by:

u∙v = ||u||×||v||×cosӨ

uӨ

v

SCALAR PRODUCT1. When vectors u and v are orthogonal,

their scalar product is ZERO

2. u∙u = ||u||×||u||×cos0˚

= ||u||2

Example 2Consider u and v form a 30˚ angle and that

||u||=4 and ||v||=3. Find the scalar product.

u∙v = 4 x 3 x cos30

u∙v = 10.39

SCALAR PRODUCT IN THE CARTESIAN PLANE

Let u=(a,b) and v=(c,d), we have:

u∙v = ac + bd

If we know the angle Ө formed by vectors u and v, we have:

cosӨ = u∙v _

||u|| ||v||

Example 3Let u=(4,1) and v=(2,3). Find the scalar product and the

angle between the vectors

u∙v = 4x2 + 1x3

u∙v = 11

What are ||u|| and ||v||?

||u||=√42+12 = √17

||v|| = √22+32 = √13cosӨ = u∙v _

||u|| ||v||

Example 3Let u=(4,1) and v=(2,3). Find the scalar product and the angle between the vectors

cosӨ = u∙v _

||u|| ||v||

cosӨ = 11 _ = 11 = 0.74

√17 x √13 √221

So Ө = cos-1(0.74)

Ө = 42.3˚

PROPERTIES OF THE SCALAR PRODUCT

u∙v = v∙u

ru∙sv = (rs)u∙v

u∙(v+w) = u∙v + u∙w

Last homework questions?(not the quiz questions…)

HOMEWORK

p. 300 #25, 26

p. 301 #27, 28, 30

p. 303 #1,2,3p. 304 #6,7,8,9 (how do you know if two vectors are perpendicular? Scalar product

is 0!)

p.306-307 try them all TAKE HOME QUIZ

DUE TOMORROW