Block 3
Dot ProductCalculating Angle
What is to be learned?
• How to use dot product to calculate the angle between vectors
From Before
a.b = |a| |b| cosθ
cosθ =
cosθ =
a.b |a| |b|
also a.b = x1x2 + y1y2 + z1z2
x1x2 + y1y2 + z1z2
|a| |b|
cosθ = x1x2 + y1y2 + z1z2
|a| |b|
Find angle between a and b214( ) 7
26( )
Numerator: 2(7) + 1(2) + 4(6) = 40
Denom: |a| = √(22 + 12 + 42) = √21
|b| = √(72 + 22 + 62) = √89
cosθ = 40
√21√89= 0.925 θ = 22.30
cosθ = x1x2 + y1y2 + z1z2
|a| |b|
Find angle between a and b213( ) -1
5 2( )
Numerator: 2(-1) + 1(5) + 3(2) = 9
Denom: |a| = √(22 + 12 + 32) = √14
|b| = √((-1)2 + 52 + 22) = √30
cosθ = 9
√14√30= 0.439 θ = 64.00
Calculating Angles with Dot Product
Rearranging formula
cosθ = a.b |a| |b|
x1x2 + y1y2 + z1z2
|a| |b|=
a = 3i + 4k ,b = 4i + 6j + 2k
cosθ = x1x2 + y1y2 + z1z2
|a| |b|
Calculate angle between vectors
Numerator: 3(4) + 0(6) + 4(2) = 20
Denom: |a| = √(32 + 02 + 42) = √25 = 5
|b| = √(42 + 62 + 22) = √56
cosθ = 20
5√56= 0.5345 θ = 57.70
cosθ = x1x2 + y1y2 + z1z2
|a| |b|
Find angle between a and b412( ) -3
4 5( )
Numerator: 4(-3) + 1(4) + 2(5) = 2
Denom: |a| = √(42 + 12 + 22) = √21
|b| = √((-3)2 + 42 + 52) = √50
cosθ = 2
√21√50= 0.0617 θ = 86.50
Key Question
If vectors are perpendicular cosθ = cos900 =
cosθ = x1x2 + y1y2 + z1z2
|a| |b|
0
x1x2 + y1y2 + z1z2 = 0
Numerator
Denominator
Perpendicular Vectors
Numerator = 0
θ
Prove a = 3i – 2j is perpendicular to
b = 4i + 6j – 5k
cosθ = x1x2 + y1y2 + z1z2
|a| |b|
If perpendicular cosθ = cos900 = 0i.e. num = 0
num = 3(4) + (-2)6 + 0(-5)
= 0 as required
must equal zero
4 -1 -2 ( )32
5( )Prove a and b are perpendicular
a = b =
If perpendicular cosθ = cos900 = 0i.e. num = 0
num = 4(3) + (-1)2 + (-2)(5)
= 0 as required
2 -1 -2( )
42 k( )
If a and b are perpendicular, find value of k
a = b =
If perpendicular cosθ = cos900 = 0i.e. num = 0
num = 2(4) + (-1)2 + (-2)(k)
= 6 – 2k
6 – 2k = 0 k = 3
If vectors are perpendicular cosθ = cos900 =
cosθ = x1x2 + y1y2 + z1z2
|a| |b|
0
x1x2 + y1y2 + z1z2 = 0
Numerator
Denominator
Perpendicular Vectors
Numerator = 0
θ
2 -1 -3
( ) 34 k( )
If a and b are perpendicular, find value of k
a = b =
If perpendicular cosθ = cos900 = 0i.e. num = 0
num = 2(3) + (-1)4 + (-3)(k)
= 2 – 3k
2 – 3k = 0 k = 2/3
7 g -9
( ) 0g 4( )
If a and b are perpendicular, find possible values of g
a = b =
If perpendicular cosθ = cos900 = 0i.e. num = 0num = 7(0) + g(g) + (-9)(4)
= g2 – 36
g2 – 36 = 0 g2 = 36 g = 6 or -6
Key Question
A (7,5 ,7) B (3 ,4 ,6) C (5,6,9)Calculate LABC
Need BA and BC
u = BA = a – b
A
B
C
θu
vu v
757 ( ) 3
46 ( )= –
411= ( )
v = BC = c – b
569 ( ) 3
46 ( )= –
223=
( )
cosθ = x1x2 + y1y2 + z1z2
|a| |b|
Angle between u and v411( ) 2
2 3 ( )
Numerator: 4(2) + 1(2) + 1(3) = 13
Denom: |a| = √(42 + 12 + 12) = √18
|b| = √(22 + 22 + 32) = √17
cosθ = 13
√18√17θ = 420