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Block 3
Vector Multiplication
What is to be learned?
• How to “multiply” vectors using the dot product(both ways!)
Eggs
Plan View
Effectiveness depends on • Strength• Direction• Here vectors are parallel (angle “between” them is zero)
( Magnitude of vector )(Angle “between” vectors)
Larger angle → Less effective
The Dot Product
Vector “Multiplication” depends onMagnitude andAngle between vectors
Most effective when vectors are parallelAngle = 00
Formula 1
a.b = |a| |b| cosθ
Max value of cosθ is 1 when θ = 0
a
b
a.b = |a| |b| cosθ7
4
600 = 7(4) cos600
= 28 X 1/2
= 14
a number!!!!
vectors start from same place
a
b
a.b = |a| |b| cosθ5
2
450 = 5(2) cos450
= 10 X 1/√2
= 10/√2
Vector Multiplication (The Dot Product)
Formula 1
a.b = |a| |b| cosθ
a
b
θ0
vectors start from same place
a
b
a.b = |a| |b| cosθ8
5
450 = 8(5) cos450
= 40 X 1/√2
= 40/√2
a number!!!!
a
b
a.b = |a| |b| cosθ5
4
300 = 5(4) cos300
= 20 X √3/2
= 10√3
Key Question
Calculate a.b
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
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
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
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
Key Question
a = 3i + 2k ,b = 4i + 5j + 3k
a.b = 3(4) + 0(5) + 2(3)
= 18
Calculate a.b
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
