Pre-Calculus Honors
Book Reference 6.2
Unit 9 Lesson 3: More Angles & Intro to the Dot Product
Objective: _______________________________________________________
1. Group Practice: Finding the angle between two vectors using the dot product
Definition & Formulas
If
q
is the angle between the non zero vectors u and v
then cos 


uv
1 u  v

and   cos 

u
v
u v


Algebraic
Directions: Find the angle between vectors u and v
u = 6i + 8j
v = 5i + 12j
In order to use this formula you need to know the
definition of the dot product u∙v
u =< u1, u2 > and
v =< v1, v2 > is u·v = u1v1 + u2 v2
The dot product of two vectors
Verbal
Directions: Write in your own words how to find the
angle in between two vectors.
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
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____________________________________________
____________________________________________
____________________________________________
____________________________________________
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Analytical
Directions: Use another method to find the angle in
between two vectors.
2. Group Investigation: Deriving the formula for finding an angle between
 uv
 u v
1
two vectors   cos 
.




a.) Explain in your own words why the distance between the vectors u and v
is ‖𝑢 − 𝑣‖.
__________________________________________________
___________________________________________________
b.) Using the law of cosines, find ‖𝑢 − 𝑣‖.
_________________2 = __________2 + __________2 − 2 __________ ⋅ _________𝑐𝑜𝑠𝜃
c.) Isolate 2‖𝑢‖‖𝑣‖𝑐𝑜𝑠𝜃 in part b.
__________________ = ____________________2 + __________2 + __________2
d.) Expand ‖𝑢 − 𝑣‖2 , ‖𝑢‖2 , 𝑎𝑛𝑑 ‖𝑣‖2 to remove the magnitude signs.
2‖𝑢‖‖𝑣‖𝑐𝑜𝑠𝜃 = ______________________________________________________________
e.) Combine like terms.
2‖𝑢‖‖𝑣‖𝑐𝑜𝑠𝜃 = ______________________________________________________________
f.) Isolate 𝑐𝑜𝑠𝜃.
𝑐𝑜𝑠𝜃 = _________________________
g.) Isolate 𝜃 by using the inverse of cosine.
𝜃 = __________ (
)
Other properties of the dot product ( You will need these to complete the vector problem
set #2)
Directions: Write in your own words what each of these properties mean.
1. u∙v = v∙u
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_____________________________________________________________
2. u ∙ u = u
2
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_____________________________________________________________
3. k(u ∙ v) = (ku)∙v
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_____________________________________________________________
4. u∙(v + w) = u∙v + u∙w
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_____________________________________________________________
5. Two vectors are perpendicular if u∙v = 0 _________________________
_____________________________________________________________
6. Two vectors are parallel if u  v   u v
_________________________
_____________________________________________________________
Vector Problem Set #2
In exercises 1-6, use the vectors u = <2, 2>, v = <-3, 4>, and w = <1, -4> to find the indicated
quantities. State whether the result is a vector or a scalar.
1. u∙u
2. u - 2
3. (u∙v)w
4. (w∙u)v
5. u∙2v
6. 4u∙v
7. Which of the following three vectors are parallel and which are perpendicular?
u = <4, -6>
v = <-2, 3>
w = <9, 6>
8. If u = <5, -3> and v = <3, 7>, verify that:
(a) u∙v = v∙u
(b) 2(u∙v) = (2u)∙v
In 9-10, verify that u∙(v + w) = u∙v + u∙w for the given vectors u, v, and w.
(a) u = <-2, 5>, v = <1, 3> , and w = <-1, 2>
(b) u = <1, -4>, v = <-2, -2>, w = <1, 5>
11. Verify that the angle between u = <2, 3> and v = <1, -5> is 135 degrees
12. Find the measure of the angle between the given vectors to the nearest tenth of a degree.
(a) u = <3, -4> and v = <3, 4>
(b ) u = <1,3> and v = <-8, 5>
Vector Problem Set #2 Answer Key
1. 8 (Scalar)
2.
8 - 2 (Scalar)
3. 2 < 1, -4 > = < 2, -8 > (Vector)
4. -6 <-3, 4 > = < 18, -24 > (Vector)
5. 4 (Scalar)
6. 8 (Scalar)
7. Vector v and w are perpendicular. The dot product is zero.
Vector u and w are perpendicular. The dot product is zero.
Vector u and v are parallel.
8a.
-6
8b.
-12
12b.
76.4 degrees
9. 25
10. -13
11. 135 degrees
12a. 106.3 degrees
Link Sheet Answer Key
Definition & Formulas
If
q
is the angle between the non zero vectors u and v
then cos 


uv
1 u  v

and   cos 

u
v
u v


Algebraic
Directions: Find the angle between vectors u and v
U = 6i + 8j
V = 5i + 12j
u∙v = 6(5) + 8(12) = 126
llull = 10
llvll = 13
 126 
  14.25
130


  cos 1 
In order to use this formula you need to know the
definition of the dot product u∙v
u =< u1, u2 > and
v =< v1, v2 > is u·v = u1v1 + u2 v2
The dot product of two vectors
Verbal
Directions: Write in your own words how to find the
angle in between two vectors.
1. Find the dot product of the two vectors.
2. Find the magnitude of vector 1.
3. Find the magnitude of vector 2.
4. Find the product of the magnitudes.
5. Find the cosine inverse of the dot product
divided by the product of the magnitudes.
Analytical
Directions: Use another method to find the angle in
between two vectors.
U = 6i + 8j
V = 5i + 12j