Download Section 11.1 – Vectors in a Plane

Section 11.1 – Vectors in a
Plane
White Board Challenge
Find the length of the segment and the
measure of the marked angle below.
y
(1, 3)
3
𝛽
1
𝜃
2
L 1  3  42
  180    180  60  120
2
x
  tan 1
   60
3
1
A Point in the Plane
The pair (𝑎, 𝑏) determines a point in the
plain.
y
(𝑎, 𝑏)
x
A Directed Line Segment
The pair (𝑎, 𝑏) also determines a directed line
segment (or arrow) with its tail at the origin
and its head at (𝑎, 𝑏).
y
(𝑎, 𝑏)
𝑎, 𝑏
x
The Magnitude of a Directed Line
Segment
The length of this arrow represents the
magnitude.
y
Magnitude = 𝑎2 + 𝑏 2
(𝑎, 𝑏)
𝑎, 𝑏
𝑏
𝑎
x
A Position Vector
The ordered pair (𝑎, 𝑏) represents a
mathematical object with both magnitude and
direction, called the position vector of (𝒂, 𝒃).
y
Magnitude = 𝑎2 + 𝑏 2
Vectors
appear in
physical
settings:
velocity
and force.
(𝑎, 𝑏)
𝑎, 𝑏
x
Notation for a Vector
A two-dimensional vector v is an ordered pair of real
numbers denoted in component form as 𝑎, 𝑏 . The
numbers 𝑎 and 𝑏 are the components of vector v.
Textbooks/Tests use
y
various symbols for
vectors:
𝑃
j
Boldface variables: v
Arrows: 𝑣
i
Angled Brackets: 𝑎, 𝑏
v = 𝑎, 𝑏
Between Points: 𝑂𝑃
Between Points: 𝑂𝑃
Component Form: 𝑎𝒊 + 𝑏j
𝑂
x
Standard Representation of a Vector
The standard representation of the vector
𝒂, 𝒃 is the arrow from the origin to the point
(𝑎, 𝑏).
y
We assume
all vectors
are based
at the
origin
unless
otherwise
stated.
(𝑎, 𝑏)
𝑎, 𝑏
x
Magnitude of a Vector
The magnitude of the vector 𝐯 = 𝑎, 𝑏 ,
denoted 𝒗 , is the length of the arrow. The
magnitude is the non-negative real number:
𝒂, 𝒃 =
2
𝑎
+
2
𝑏
The magnitude is sometimes referred to as the
“absolute value” of the vector OR symbolically as
𝒗 = 𝑎, 𝑏 (Our book uses the double bar notation).
Direction Angle of a Vector
The direction of the vector 𝐯 = 𝑎, 𝑏 is the direction in
which the arrow is pointing.
The direction angle of a nonzero vector 𝐯 = 𝑎, 𝑏 is the
angle 𝜃 formed with the positive x-axis as the initial ray and
the standard representation of 𝐯 as the terminal ray
(measured counter-clockwise).
Create a right triangle and use inverse
y
𝑎, 𝑏
tangent along with
supplementary/complimentary/360°
angles to find 𝜽.
𝜃
x
EVERY BOOK DEFINES THIS ANGLE
DIFFERENTLY
Equivalent Vectors
It is often convenient to represent vectors with
arrows that begin at points other than the origin.
Two arrows with the same length and pointing in the
same direction represent the same vector. Two
arrows that represent the same vector are said to be
equivalent.
Example: 𝑂𝑃 and 𝑄𝑅 are equivalent vectors.
𝑅(−2,6)
6
5
𝑃(2,4)
4
3
𝑄(−4,2)
2
1
-4
-3
-2
-1
𝑂
1
2
3
White Board Challenge
A vector has an initial point (−1,4) and
terminal point (2,1). Find the magnitude and
direction angle of the vector.
(−1,4)
Find the “Standard” Vector:
2  1,1  4  3, 3
(2,1)
𝜃
𝛽
Find the length of the “Standard” Vector:
3, 3  3   3  18  3 2
2
2
3
3
Find the direction angle of the “Standard” Vector:
  tan
1
 33   45
  360    360  45  315
The Sum of Vectors
Let 𝒖 = 𝑢1 , 𝑢2 and 𝒗 = 𝑣1 , 𝑣2 be vectors and let 𝑘 be a
real number (scalar).
The sum (or resultant) of the vectors 𝒖 and 𝒗 is the
vector
𝒖 + 𝒗 = 𝑢1 + 𝑣1 , 𝑢2 + 𝑣2
Add the corresponding components.
Example: Find 4,3 + −2,1 and illustrate the result
graphically.
4,3  2,1  4  2,3  1
 2, 4
2,4
4,3
−2,1
Product of a Scalar and a Vector
Let 𝒖 = 𝑢1 , 𝑢2 and 𝒗 = 𝑣1 , 𝑣2 be vectors and let 𝑘 be a
real number (scalar).
The product of the scalar 𝒌 and the vector 𝒖 is the
vector
𝑘𝒖 = k 𝑢1 , 𝑢2 = 𝑘𝑢1 , 𝑘𝑢2
Multiply both of the components.
Example: Find 2 2,1 and illustrate the result graphically.
2 2,1  2  2, 2 1
 4, 2
2,1
2,4
White Board Challenge
Let 𝐮 = −1,3 and 𝐯 = 4,7 . Find the following.
(a) 2𝒖 + 3𝒗  2
1,3  3 4, 7  2, 6  12, 21
 2  12, 6  21  10, 27
(b) 𝒖 − 𝒗
 1,3  4, 7  1  4,3  7
 5, 4
(c)
1
𝒖
2

1
2
1,3 

1
2
 1, 12  3  0.5,1.5
 0.5
2
 1.5  2.5
2