Dot Product Calculation Angles

1. Block 3
Dot Product
Calculating Angle

2. What is to be learned?
• How to use dot product to calculate the
angle between vectors

3. From Before
a.b = |a| |b| cosθ
cosθ =
cosθ =
a.b
|a| |b|
also a.b = x1x2 + y1y2 + z1z2
x1x2 + y1y2 + z1z2
|a| |b|

4. cosθ = x1x2 + y1y2 + z1z2
|a| |b|
Find angle between a and b
21
4
( )
7
2
6
( )
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

5. cosθ = x1x2 + y1y2 + z1z2
|a| |b|
Find angle between a and b
21
3
( )
-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

6. Calculating Angles
with Dot Product
Rearranging formula
cosθ = a.b
|a| |b|
x1x2 + y1y2 + z1z2
|a| |b|
=

7. 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

8. cosθ = x1x2 + y1y2 + z1z2
|a| |b|
Find angle between a and b
41
2
( )
-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

9. If vectors are perpendicular
cosθ = cos900
=
cosθ =
x1x2 + y1y2 + z1z2
|a| |b|
0
x1x2 + y1y2 + z1z2 = 0
Numerator
Denominator
Perpendicular Vectors
Numerator = 0
θ

10. Prove a = 3i – 2j is perpendicular to
b = 4i + 6j – 5k
cosθ = x1x2 + y1y2 + z1z2
|a| |b|
If perpendicular cosθ = cos900
= 0
i.e. num = 0
num = 3(4) + (-2)6 + 0(-5)
= 0 as required
must equal zero

11. 4
-1
-2
( )
32
5( )
Prove a and b are perpendicular
a = b =
If perpendicular cosθ = cos900
= 0
i.e. num = 0
num = 4(3) + (-1)2 + (-2)(5)
= 0 as required

12. 2
-1
-2( )
42
k
( )
If a and b are perpendicular, find value of k
a = b =
If perpendicular cosθ = cos900
= 0
i.e. num = 0
num = 2(4) + (-1)2 + (-2)(k)
= 6 – 2k
6 – 2k = 0
k = 3

13. If vectors are perpendicular
cosθ = cos900
=
cosθ =
x1x2 + y1y2 + z1z2
|a| |b|
0
x1x2 + y1y2 + z1z2 = 0
Numerator
Denominator
Perpendicular Vectors
Numerator = 0
θ

14. 2
-1
-3
( )
34
k
( )
If a and b are perpendicular, find value of k
a = b =
If perpendicular cosθ = cos900
= 0
i.e. num = 0
num = 2(3) + (-1)4 + (-3)(k)
= 2 – 3k
2 – 3k = 0
k = 2
/3

15. 7
g
-9
( )
0g
4
( )
If a and b are perpendicular, find possible
values of g
a = b =
If perpendicular cosθ = cos900
= 0
i.e. num = 0
num = 7(0) + g(g) + (-9)(4)
= g2
– 36
g2 – 36 = 0
g2
= 36
g = 6 or -6
Key
Question

16. 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
θ
u
v
u v
75
7
( )
3
4
6
( )= –
41
1=
( )
v = BC = c – b
56
9
( )
3
4
6
( )= –
22
3=
( )

17. cosθ = x1x2 + y1y2 + z1z2
|a| |b|
Angle between u and v
41
1
( )
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√17
θ = 420