Vectors
Scalar Quantities have size (magnitude) but no direction.
Vector Quantities have both direction and size (magnitude).
How are vectors represented? Usually by a ray that goes from one point to another.
1. 2 Points: vector AB starts at A (tail) and goes to B (head)
2. A lower case letter that is bolded (book) or hand written with an arrow above it or
a line below it.
Vectors are only equal if they have the same direction and magnitude.
Negative Vectors have the same magnitude but the opposite direction.
Note: AB   BA
Example:
Draw a vector representing 40 km/h winds with a bearing of 120 degrees.
Column Vector Form (Component)
Components are the x and y parts of a vector. (change in x and change in y) The vectors
components are written like a 2x1 matrix.
Unit Vector Form:
 0  and  1  are special unit vectors because they are each one unit in length, vertical or
1 
 
0
 
horizontal and positive. i =  1  and j =  0  therefore any vector can be written in the
0
 
1 
 
form:  a1 
 a   a1i + a2 j
 2
The magnitude of vector
AB is written as: AB or a .
Example: Also find the magnitude of each.
Parallelism: If two vectors are parallel then one is a scalar multiple of the other.
If
If
a ║ b then a  kb where k is a scalar.
a  kb then a  k b and a ║ b
Remember lines are parallel if they have the same gradient (same direction) so the size
does not matter.
Example:
Example:
Find scalar j if the parallel vectors are: a  

2
7
,
b


 j
 1
 
Vector Addition: To add vectors you draw one and draw the second at the head of the
first. Finally connect the tail of the first with the head of the second. This will be called
the resultant vector.
b
a+b=c
a
c (resultant vector)
Example: Using the above diagram add two vectors to get each of the
following: b, - c, and –a
Example: Construct a vector equation for r, and p:
r
p
q
s
Example: Find the components of vector AB if A = (1, 3) and B = (4, -4)
and find the distance of AB.
Adding 2 vectors by their components
 b1 
 a1 

,
b
b 

 2
 a2 
a 
 a1  b1 

a2  b2 
a+b 
Example: Use vector methods to find the remaining vertex of parallelogram
ABCD where A (-1, 3), B (2, 4), C (-2, -1).
Example: Find the scalars m and n such that:
2
5 8
m    n    
 1
 2   3
Example:
Challenge:
Stephanie swims at 1.8m/s from X and wants to reach Y. The river flows 0.3m/s east and
the distance of YP = 20 and XP= 80:
A
Y
P
a) Find the distance of XY.
b) What direction should she head?
c) How long will it take her?
X
Reminder:
 x
 x2  x1 
 


The position vector of OP  y , and PQ  y2  y1
 


z
z z 
 
 2 1
 v1 
 
2
2
2
The magnitude of v  v2 is v  v1  v2  v3
 
v 
 3
Remember v1  x2  x1
How would we represent the midpoint formula?
Example:
Find the magnitude and midpoint of a vector that starts at A(0, -1, 2) and
goes to B (3, 2, -4)
Example:
Graph the following line segment M (1, 2, 2) and N (-2, 1, 3) and find the
following: MN and NM
Example:
A (-1, 3, 4), B(2, 5, -1), C(-1, 2, -2), and D(r, s, t) are 4 points in space. Find
r, s, and t if: AB  DC .
Example:
If A, B, C, and D are collinear in that order, all are equally spaced apart, A (2, -5, 3) and B (-1, -2, 2), find the coordinates of C and D.
 2  1  7 
    

Example: Find r, and s such that: r 3  s 7  19
    

 1   2  2 
    

2
1
 
 
If a  3 , b  1 find the following:
 
 
1
2
 
 
(a) 3a
(b) 3  2b  2a 


Unit Vectors
Unit Vectors are vectors that are 1 unit long.
Example 1: Are the following vectors unit vectors?
a=
0
 1 , b
 
1
2
= , c
1
 2 
=





3

2 
1 
2 
Example: Find the length of vector:
3.5i + 4 j
The unit vector in the direction of a: a  1 a
a
The vector
b
of length k in the same direction as
a
is b  k a
a
The vector
b
of length k which is parallel to
Example: If vector b is parallel to a where
a
is b   k a
a
 2  and k = 3 find b.
a 
 3 
Example: How can we find a vector of 10 units going in the same direction as
a  2i - 3j + k ?