vectors and applications bomla lacymath summer 2015

Vectors and ApplicationsBOMLA LacyMath

Summer 2015

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 are used to represent quantities such as force and velocity.

Vectors have both magnitude and direction.


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


Example 1

Draw a vector with initial point (3, 4) and terminal point (7, 8).


Example 2

Draw a vector with initial point (-6, 3) and terminal point (4, -2).


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!


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) >


Back to Example 1 (put in component form)

Draw a vector with initial point (3, 4) and terminal point (7, 8).



Draw a vector with initial point (-6, 3) and terminal point (4, -2).



Draw a vector with initial point (-1, -5) and terminal point (0, 3).


The magnitude of a vector is the length of the arrow.

The direction is the direction in which the arrow is pointing.


Back to Example 1

Find the magnitude of the vector.


Back to Example 2



Back to Example 3



The magnitude of a vector is…

|v| =

if given the initial and terminal points.

|v| =

if given the component form.


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?


What about direction?

What are we given when we know the component form of a vector? (Think, where does the vector start?)


Direction of a Vector

Found by using…

is the angle that the vector makes with

the positive x-axis.


Back to Example 1

Find the direction of the vector.


Back to Example 2



Back to Example 3



If you know the direction angle of the vector, the components can also be found using…

< |v|cos() , |v|sin() >


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 >


Back to Example 1

Write this vector as a combination of standard unit vectors.


Back to Example 2



Back to Example 3



Vector Operations

Vectors can be added and subtracted by using their components.


Vector Operations

Example 1

v = < 3, 6 > and u = < 7, 10 >

Find u + v, u – v, and 3u + 2v.


CWPg. 511#5 – 8, 13, 15 – 17, 29, 30

vectors and applications bomla lacymath summer 2015

Documents

applications vectors

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vertical vectors

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vertical component vectors

applicationsany vector

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