Transcript Head to Tail Vector Addition

Scalar Quantities
• A scalar quantity (measurement) is one
that can be described with a single
number (including units) giving its size or
magnitude
• Examples: Temperature, time, mass,
distance.
• Can you think of some?
Vectors
• A vector quantity is one that inherently
requires magnitude and direction.
• Examples: Force, acceleration,
displacement.
• Can you think of some?
Representing vectors
• When describing a vector it is
useful to draw them.
• An arrow is used to create a
graphical representation.
• The length of the arrow is
proportional to the magnitude.
• The direction of the arrow
corresponds to the direction of the
vector.
Head to Tail Vector Addition
If you have to add vectors you can’t just add the
numbers to get the result. The reason is they are
pointing in different directions
6N
6 N + 8 N = 14 N
8N
Adding Vectors
To add them we can simply move one so the head
of one and tail of the other match up. This has to
be done or our result will not be the right magnitude
6N
Resultant
Measure angle
8N
Copy length
Copy angle
Analytical addition
• Although graphical methods
work they require very precise
drawings
• The analytical method uses
trigonometry to add the
vectors mathematically
• Lets review triangles
Meaning of sine, cosine & tangent
• The ratio of two sides of any right triangle with
the same interior angles is always the same
number independent of the size of the triangle,
the triangle's orientation, or the units used to
measure the sides of the triangles.
• There are three ways to form a ratio of the three
sides of a triangle (six if you count the inverse
ratios also).
• The symbols "sin", "cos", and "tan" are the
tags/labels we use to identify which ratio is
which.
Example
• Imagine a right
triangle with the
interior angles of 30,
60 & 90
• Let’s look at the ratio
of sides
• The sine would be ½
• Click here for the unit
circle
2
1
30
√3
The Ratio
• The ratios are programmed into the
calculator, and It’s really just using the unit
circle
Co-linear and Perpendicular
• Co-linear vectors are on the same axis or
number line
• Perpendicular vector simply make right angle
with each other
Adding Perpendicular Vectors
• You can see by closing
the tip of one and the
tail of the other it makes
a triangle
• For this we use the
Pythagorean theorem
• The formula a2 + b2 = c2
relates the three sides
of a triangle
Vector components
• Sometimes vectors are not co-linear (on
the same number line), or perpendicular
• For this we must find its “components”
• These are the x and y pieces that make up
the right triangle using the vector as the
hypotenuse
What?
• Look at the following vector.
• Imagine we shine a light from
directly above it facing down.
• It will cast a shadow on the x axis
equal to its length in the x direction
That shadow is the x-component
What?
• Look at the same vector but now for the y-direction.
• Imagine we shine a light from the east side facing right.
• It will cast a shadow on the y axis equal to its length in the
y direction
That shadow is
the y-component
Unit circle