This document discusses using the dot product to calculate the angle between two vectors. It provides the formula for calculating the dot product and shows examples of using the formula to find the angle between vectors. It also discusses how if two vectors are perpendicular, their dot product will equal 0.

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