Vector Multiplication and Dot Products

1. Block 3
Vector Multiplication

2. What is to be learned?
• How to “multiply” vectors using the dot
product
(both ways!)

3. Eggs

4. Plan View
Effectiveness depends on
• Strength
• Direction
• Here vectors are parallel
(angle “between” them is zero)
( Magnitude of vector )
(Angle “between” vectors)

5. Larger angle → Less effective

6. The Dot Product
Vector “Multiplication” depends on
Magnitude and
Angle between vectors
Most effective when vectors are parallel
Angle = 00

7. Formula 1
a.b = |a| |b| cosθ
Max value of cosθ
is 1 when θ = 0

8. a
b
a.b = |a| |b| cosθ
7
4
600 = 7(4) cos600
= 28 X 1
/2
= 14
a number!!!!
vectors start from same place

9. a
b
a.b = |a| |b| cosθ
5
2
450 = 5(2) cos450
= 10 X 1
/√2
= 10
/√2

10. Vector Multiplication
(The Dot Product)
Formula 1
a.b = |a| |b| cosθ
a
b
θ0
vectors start from
same place

11. a
b
a.b = |a| |b| cosθ
8
5
450 = 8(5) cos450
= 40 X 1
/√2
= 40
/√2
a number!!!!

12. a
b
a.b = |a| |b| cosθ
5
4
300 = 5(4) cos300
= 20 X √3
/2
= 10√3
Key Question
Calculate a.b

13. Dot Product 2!
a.b = x1x2 + y1y2 + z1z2
( )
x1
y1
z1
( )
x2
y2
z2
a = b =

( )
2
4
6 ( )
3
5
7
a = b =
a.b = 2(3) + 4(5) + 6(7)
= 68

14. Dot Product 2!
a.b = x1x2 + y1y2 + z1z2
( )
x1
y1
z1
( )
x2
y2
z2
a = b =

( )
3
-2
0 ( )
-2
-4
7
a = b =
a.b = 3(-2) + (-2)(-4) + 0(7)
= 2

15. Dot Product 2!
a.b = x1x2 + y1y2 + z1z2
( )
x1
y1
z1
( )
x2
y2
z2
a = b =

a = 2i – 3j + k b = 4j – k
a.b = 2(0) + (-3)(4) + 1(-1)
= -13

16. Formula 2
a.b = x1x2 + y1y2 + z1z2
( )
x1
y1
z1
( )
x2
y2
z2
a = b =

( )
5
2
3 ( )
-8
5
1
a = b =
a.b = 5(-8) + 2(5) + 3(1)
= -27

17. Key Question
a = 3i + 2k ,b = 4i + 5j + 3k
a.b = 3(4) + 0(5) + 2(3)
= 18
Calculate a.b

18. a
b
a.b = |a| |b| cosθ
7
4
600
= 7(4) cos1200
= 28 X -1
/2
= -14
b starts here
a starts hereboth start
here
1200