Vectors and Dot Product
6.4
JMerrill, 2010
Quick Review of Vectors: Definitions
• Vectors are quantities that are described by
direction and magnitude (size).
• Example: A force is a vector because in order to
describe a force, you must specify the direction in
which it acts and its strength.
• Example: The velocity of an airplane is a vector
because velocity must be described by direction
and speed.
• The vector 0, 0 is the zero vector. It has no
direction.
Representation of Vectors
• The velocities of 3 airplanes, two of which are
heading northeast at 700 knots, are represented
by u and v
u
v
We say u = v to
indicate both planes
have the same
velocity.
w
w ≠ u or v because the direction is different.
Magnitude
• The magnitude of vector v is represented by the
absolute value of v (with double bars)
• In the previous example, u  700, v  700, w  700
We know that u  v  w , but u ≠ w, and v ≠ w
because the direction is not the same.
Addition of Vectors
v  AB
AB  BC  AC
AC is the vector sum
of AB  BC
The sum is called
the resultant.
A
10
B
5
C
Component Form
• From the tail to the tip of
vector v, we see:
• A 2 unit change in the xdirection, and
• A -3 unit change in the ydirection.
• 2 and -3 are the components
of v.
• When we write v = 2, -3 ,
we are expressing v in
component form.
2
3
Component Form
• You can count the number of
spaces to get the component
form or, you can subtract
the coordinates.
(x2,y2)
AB  x2  x1 , y2  y1
(y2 – y1)
• IT IS ALWAYS B – A!
• The magnitude of vector AB
is found using the distance
formula:
AB  ( x2  x1 ) 2  ( y2  y1 ) 2
(x1,y1)
(x2 – x1)
Example
• Given A(4, 2) and B(9, -1), express AB in component
form. Find AB
AB  9  4, 1  2  5, 3
AB  52  ( 3)2  34
Vector Operations with Coordinates
 Vector Addition
v + u =
a,b + c,d = a+c, b+d
 Vector Subtraction
v - u =
a,b - c,d = a-c, b-d
 Scalar Multiplication
 kv =
k a,b = ka, kb
Example
• If u = 1, 3 and v = 2,5 , find:
• u+v
1+2, -3+5 = 3, 2
• u–v
1 - 2, -3 - 5 = -1, -8
• 2u – 3v
2 1, -3 - 3 2, 5 = 2, -6 - 6, 15 = -4, -21
Definition – Dot Product
Example: Find the dot product:
6 (-2) + (1)(3) = -12 + 3 = -9
The dot product gives you a
scalar, NOT a vector!
6,1  -2, 3
Properties of the Dot Product
• Let u, v, and w be vectors in the plane or in space
and let c be a scalar.
1. u  v  v  u
2. 0  v  0
3. u  (v  w)  u  v  u  w
4. v  v  v
2
5. c(u  v)  cu  v  u  cv
Angle Between Two Vectors
Example
• Find the angle between
cos  
1
1,5 and -2,4 .
(1)( 2)  (5)( 4)
1 5
2
cos  38
2
(2)  4
2
2
 .789
Orthogonal (Perpendicular) Vectors
• Two vectors are orthogonal if their dot product is 0
uv  0
• Example:
Let u  2, 3 and v  3,2
u  v  (2)(3)  ( 3)(2)  0
so these vectors are othogonal
You Try
• Find 4,5  2,3
• Find u  v  w
23
14,28
• Find the angle between u  4,3 and v  3,5
 22.2o