Mathematical Investigations IV
Name:
Vectors
Getting To the Point
Dot Products Made Easy
(Instructions included)
Recall the diagram from the previous packet. You found
that in vector terms, x = v – w. If v has components
<v1, v2> and w = <w1, w2>, what are the components of x?
x=<
w

>
,
Write expressions for the squares of the magnitudes of v,
w, and x in terms of their coordinates: v1, v2, w1, and w2.
v 2
x
v
x 2
w 2
Apply the Law of Cosines to this triangle, substituting the expressions you found for |v|2, |w|2,
and |x|2. Expand, and simplify:
|x|2 = |v|2 + |w|2 – 2|v||w|cos()
Now solve for |v| |w| cos.
This intriguing formula relating the geometry and algebra of vectors is so important that we give
the algebraic expression its own name.
Definition of the Dot Product
vw = v1w1 + v2w2
As a consequence of our work, we have shown:
Geometrical Interpretation of the Dot Product
vw = |v| |w| cos,
where  is the angle between v and w.
Vectors 5.1
Rev F09
Mathematical Investigations IV
Name:
Now let’s try some problems:
1.
If v = <–2, 6> and w = <3, 4>, find vw.
2.
Find <1,1>  <3,–5>.
3.
Find a so that <2, 6> and <–2, a> are orthogonal (perpendicular).
4.
Use the dot product in reverse to find the angle between <3, 2> and <4, 1>.
MORE PRACTICE
1.
Find v1 . v2. Then, determine if the angle between v1 and v2 is acute, right, or obtuse (or
possibly none of these).
a.
v1 =  2, 5  and v2 =  –3, 1
b.
v1 = –11 iˆ – 3 ĵ
c.
v1 =  4, 9  and v2 =  –18, 8 
d.
2 and v = 6  [Be careful. The answer is not acute.]
v1 = 
2 12

4

 





and v2 = –2 iˆ – 1 ĵ
Vectors 5.2
Rev F09
Mathematical Investigations IV
Name:
2.
Two non-zero vectors v1 and v2 are parallel if there is k  0 so that k v1 = v2.
a.
Find a so that  3, 5  and  a + 3, 20  are parallel.
b.
3.
Explain why  –4, 6  and  8, –12  are parallel.
Determine the angle between the following vectors.
a.
 2, 5  and  8, 20 
b.
 –2, 7  and  6, –21 
c.
 2, 7  and  –4, 5 
d.
 –6, 3  and  4, 0 
4.
Let v1 =  2t + 3, 5  and v2 =  12, t – 2 . Find t so that v1 · v2 = 0.
5.
Let v1 =  2z + 3, 5  and v2 =  z – 2, –12 .
a.
Find z so that v1 · v2 = 0.
b.
Find z so that v1 · v2 is a minimum.
Vectors 5.3
Rev F09
Mathematical Investigations IV
Name:
6.
a. Writing v  x, y , consider the equation.
v  2,1  0.
Graph all vectors v which satisfy this equation.
b. Now graph all vectors v which satisfy the equation v  2,1  4 on the same graph.
Do the same with the equation v  2,1  3.
c. What do you conclude? If you had to graph the equation v  a  k, what important
characteristic of this line would be described by the vector a? How would this
characteristic be described? (Hint: Refer to a as the vector <a1, a2> ).
Vectors 5.4
Rev F09