Mathematical Investigations IV
Name:
Vectors
Getting To the Point
Vectors - Large and Small
We’ve added, and now it’s time to look at multiplication. Multiplication of real numbers is
different from multiplication of matrices, for example, and multiplication of vectors also needs to
be defined.
Multiplication as applied to vectors has a number of incarnations.




multiplication of a vector by a scalar
the “dot (or inner) product”
multiplication of a vector by a matrix
the “cross (or vector) product”
Let’s investigate the first of these versions of vector multiplication.
Multiplication of a vector by a scalar
What is a scalar?
* a scalar is a number
* a scalar has size but not direction or geometry
Scalar Multiplication
For a scalar c and a vector v   a ,b 
cv   ca ,cb 
Multiplication by a scalar changes the length of a vector. If the scalar is negative, the direction
is reversed. For example:
v
3v
–v/2
Vectors 3.1
Rev. F05

Mathematical Investigations IV
Name:
 3
7 , find the following:
For v   and w  

2
 1 

 
1.
v
2
2.
v
3.
1
 w
7
4.
3w
5.
20v  5w
6.
2w  3v
7.
In the diagram below, A and B are midpoints of PQ and QR , respectively.
Q
v
A
e.
Express AB in terms of v and w.
b.
Express PR in terms of v and w.
c.
According to your answers above, state the
relationship between the segments AB and
PR.
w
B
R
P
8.
a.
In the diagram below, ABCD is a parallelogram, F is the midpoint of AD , and
1
1
1
AG = AB. Express each of the following in terms of u = AD and v = AB :
3
2
3
AE =
f.
a.
AD =
b.
AB =
c.
AC =
d.
DC =
BE =
Vectors 3.2
g.
GC =
Rev. F05
Mathematical Investigations IV
Name:
9.
1
4
AC, AF = AD, and
3
5
D is the midpoint of CE . Express each of the
following in terms of v and w.
In the figure at the left, AB =
C
w
D
B
F
v
A
10.
ED =
AE =
AF =
DF =
BF =
E
Given that v = <2, 4> and w = <–3, 1>, draw the following vectors.
a.

AC =
b.
2v + 3w
1
v – 3w
2



11.

Find scalars a and b such that v = ax + by where v = <5, –3>,
x = <–4, 1>, and y = <3, –1>.

Vectors 3.3
Rev. F05