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Vectors and Applications
This unit covers the topic of vectors, which loosely falls in line with the topic of right triangle trigonometry. These topics are important in many real-world applications, such as calculating the path of the wind on an airplane’s path.
Vectors and Applications
Vectors are used to represent quantities such as force and velocity.
Vectors have both magnitude and direction.
Vectors and Applications
Vectors are represented in the coordinate plane using an arrow. These arrows are drawn using an initial point and a terminal point.
Initial point: Where the vector begins Terminal point: Where the vector
ends
Vectors and Applications
Example 1
Draw a vector with initial point (3, 4) and terminal point (7, 8).
Vectors and Applications
Example 2
Draw a vector with initial point (-6, 3) and terminal point (4, -2).
Vectors and Applications
Example 3
Draw a vector with initial point (-1, -5) and terminal point (0, 3).
Vectors and Applications Any vector that has an initial point at the origin is
said to be in standard position.
A two-dimensional vector v is an ordered pair of real numbers, and is represented by its component form given by < a, b >.
“a” is the horizontal component
“b” is the vertical component
Vectors in component form start at the origin!
Vectors and Applications
To put a vector in component form, use the “Head Minus Tail” method.
(Terminal Point) – (Initial Point)
Initial Point: (x1, y1)
Terminal Point: (x2, y2)
< (x2 – x1) , (y2 – y1) >
Vectors and Applications
Back to Example 1 (put in component form)
Draw a vector with initial point (3, 4) and terminal point (7, 8).
Vectors and Applications
Back to Example 2 (put in component form)
Draw a vector with initial point (-6, 3) and terminal point (4, -2).
Vectors and Applications
Back to Example 3 (put in component form)
Draw a vector with initial point (-1, -5) and terminal point (0, 3).
Vectors and Applications
The magnitude of a vector is the length of the arrow.
The direction is the direction in which the arrow is pointing.
Vectors and Applications
The magnitude of a vector is…
|v| =
if given the initial and terminal points.
|v| =
if given the component form.
Vectors and Applications
What about direction?
The direction of the vector is the angle that the vector makes with the positive x-axis.
But how do we find it?
Vectors and Applications
What about direction?
What are we given when we know the component form of a vector? (Think, where does the vector start?)
Vectors and Applications
Direction of a Vector
Found by using…
is the angle that the vector makes with
the positive x-axis.
Vectors and Applications
If you know the direction angle of the vector, the components can also be found using…
< |v|cos() , |v|sin() >
Vectors and Applications
Standard Unit Vectors
These are the simplest forms of horizontal and vertical vectors that can be created. They are represented by i and j. Any vector can be written as a combination of these vectors.
i = < 1, 0 >
j = < 0 , 1 >
Vectors and Applications
Back to Example 1
Write this vector as a combination of standard unit vectors.
Vectors and Applications
Back to Example 2
Write this vector as a combination of standard unit vectors.
Vectors and Applications
Back to Example 3
Write this vector as a combination of standard unit vectors.
Vectors and Applications
Vector Operations
Vectors can be added and subtracted by using their components.
Vectors and Applications
Vector Operations
Example 1
v = < 3, 6 > and u = < 7, 10 >
Find u + v, u – v, and 3u + 2v.