Geometry
Name______________________
Vectors II: The Dot Product & Angle Between Vectors
April 25/26
Introduction:
Given our previous study of vectors, we saw that adding (or subtracting) two vectors results in
another vector.
For example, given:
u  2, 4
and
v  3, 8 .
Draw a parallelogram using these vectors (use the origin as a vertex) on the grid below:
Find u + v (in component form) and find
how it relates with the parallelogram.
u+v 
7
6
5
4
3
Find u  v (in component form) and find
how it relates with the parallelogram.
2
1
uv 
2
4
6
8
Multiplying Vectors: The Dot Product
Unlike scalars, there are actually two different “products” of vectors; the cross-product and the
dot product. For the purpose of this course, we will focus on the dot product (and applications).
Definition: The dot product of vectors is…
u  v  u x , u y  vx , v y  u x vx  u y v y
**note that the dot product is a scalar**
Now, a quick practice with the dot product
u  2, 3
Suppose we are given:
and
Find u  v :
(note: you should get 8 as your answer)
v  1,  2 .
10
Geometry
Find the following dot products:
(a ) 2,3  4,5 
(b) 4,5  2,3 
(c) 4,5  1, 2 
(d ) 4,5  1,5 
(e) 4,5  5, 4 
( f ) 1, 2  2,3 
( g ) 2, 4  2,3 
(h) 1, 2  4, 6 
(i ) 1, 2  1, 2 
From your answers to the above problems answer the following?
(a) Is the dot product commutative? study (a) & (b)
(b) Is the dot product distributive over addition? study (b), (c), & (d)
(c) If a vector is multiplied by a scalar, what happens to the dot product? study (f), (g), & (h)
(d) What do you notice about the two vectors in (e) and their dot product?
(e) What do you notice about the dot product of a vector and itself with respect to its
magnitude? study (i)
Geometry
To summarize…
Properties of Dot Products:
In the table below you will find a summary of the properties of vectors you just finished
exploring:
What does this last property mean?
the dot product of a vector and itself is equal to
________________________________
Prove the last property using u  x, y
Use properties of vectors to expand the following:
(a ) (u  v )  (u  v )
(b) (u  v )  (u  v )
(c) (u  v )  (u  v )
Geometry
Q: How can we find the distance between two points using vectors?
8
6
A
4
B
2
O
-5
5
-2
If   AOB , write (but do not solve) an equation using the three sides of the triangle and the
angle  for AOB -- hint: Law of Cosines
If A is the point ( xa , ya ) , then the vector OA is u  xa , ya and B is the point ( xb , yb ) with
vector OB v  xb , yb . Mark the vector u  v on the diagram.
What is an expression for the length of AB in vector AND component form?
_________________________________
Vector form (in terms of u and v )
____________________________________
Component Form
Geometry
Angle between two Vectors:
While it is relatively uncomplicated determining the direction of a vector with respect to the
horizontal, it is less obvious how to determine the angle between two vectors (see below):
14
12
10
8
u
6
4
v
2

5
10
15
20
Consider the following when thinking of a “strategy” to obtain  :



we can calculate the length (magnitude) of each vector
we can find the vector connecting the terminal ends of each vector (draw it in), which
implies we can also find its length (we can use a dot product to express this length in
terms of the two vectors)
with a triangle, three given lengths, and an angle  we can write an equation using the
Law of Cosines (as we did on the previous page).
Write this equation using vectors (but do not solve):
We could expand our equation to prove: cos  
uv
u v
** you should challenge yourself to prove this! **
Suppose we are told that two vectors are “orthogonal” (i.e. perpendicular)…
What should be true of the dot product, u  v , given cos  
uv
? _______________________
u v