Download Non Right Triangle Vector Addition

Non Right Triangle Vector Addition
Sections 1.7 and 1.8
Question:
Why in the name of all that is good would
someone want to do something like THAT?
Answer:
Because there is no law that states vectors
must add up to make right triangles.
(Oh, but if only there were.)
CONSIDER THE FOLLOWING...
An ant walks 2.00 m 25° N of E , then turns and walks 4.00 m
20° E of N.
The total displacement of the ant…
dt
4.00 m
2.00 m
…can not be found using right-triangle math because WE DON’T
HAVE A RIGHT TRIANGLE!
We can add the two individual
displacement vectors together
by first separating them into
pieces, called x- & y-components
Into WHAT?????????
COMPONENTS
Every vector can be thought of as pointing
somewhat horizontally….
[This is the black
vector’s shadow on
the y-axis]
[This is the black vector’s shadow on the x-axis]
…and somewhat vertically.
They’re kind of like the vector’s shadows.
If we add the x- and y-components
together…
they create the original vector…
…and it makes a right triangle!
Just a few
things to keep
in mind...
Since X-component vectors can point either EAST or
WEST…
EAST is considered positive.
WEST is considered negative.
Who's Law Is It, Anyway?
 Murphy's Law:
 Anything that can possibly go wrong, will go wrong (at the
worst possible moment).
 Cole's Law ??
 Finely chopped cabbage
8
Law of Sines
A
Let ∆ABC be any triangle with a, b and c
representing the measures of the sides
opposite the angles with measures A, B,
and C respectively. Then
b
c
B
C
a
sin A
––––––
a
=
sin B
––––––
b
=
sin C
––––––
c
Law of Sines can be used to find missing parts of triangles that are not right
triangles
Case 1: measures of two angles and any side of the triangle (AAS or ASA)
Case 2: measures of two sides and an angle opposite one of the known sides of
the triangle (SSA)
The Law Cosines
 Now use it to solve the triangle below.
 Label sides
and angles
C
15
 Side c first
=26°
A
12.5
B
c
c  b  a  2  a  b  cos C
2
2
2
c  15  12.5  2 15 12.5  cos 26
2
10
2
Applying the Cosine Law
c  15 2  12.5 2  15  12.5  cos 26 o
 C = 6.65
C
15
26°
A
 Now calculate the angles using the Law of Sines.
c = 6.65
sin A
––––––
12.5
11
=
sin B
––––––
15
=
sin 260
––––––
6.65
12.5
B
Wonder Woman Jet Problem
Suppose Wonder Woman is flying her invisible jet. Her
onboard controls display a velocity of 304 mph 10 E of N.
A wind blows at 195 mph in the direction of 32 N of E.
What is her velocity with respect to Aqua Man, who is
resting poolside down on the ground?
vWA = velocity of Wonder Woman with respect to the air
vAG = velocity of the air with respect to the ground
(and Aqua Man)
vWG = velocity of Wonder Woman with respect to the ground
(and Aqua Man)
We know the first two vectors; we need
to find the third. First we’ll find it using
the laws of sines & cosines, then we’ll
check the result using components.
Either way, we need to make a vector
diagram.
32
80
32
100
vWG
vWG
10
vWA + vAG = vWG
80
The 80 angle at the lower right is the complement of the 10 angle.
The two 80 angles are alternate interior. The 100 angle is the
supplement of the 80 angle. Now we know the angle between red
and blue is 132.
The law of cosines says: v2 = (304)2 + (195)2 - 2 (304) (195) cos
132 So, v = 458 mph. Note that the last term above appears
negative, but it’s really positive, since cos 132 < 0. The law of
sines says:
sin 132
v
=
sin 
195
So, sin  = 195 sin 132 / 458, and   18.45
132
v
This means the angle between green and
the horizontal is 80 - 18.45  61.6
Therefore, from Aqua Man’s perspective, Wonder
Woman is flying at 458 mph at 61.6 N of E.

80
Wonder Woman Problem: Component Method
This time we’ll add vectors via components as we’ve done before.
Note that because of the angles given here, we use cosine for the
vertical comp. of red but sine vertical comp. of blue. All units are mph.
103.3343
32
165.3694
299.3816
10
52.789
Combine vertical & horiz. comps. separately and use Pythagorian
theorem.
 = tan-1 (218.1584 / 402.7159) = 28.4452.  is measured from the
vertical, which is why it’s 10 more than  was.
165.3694
103.3343
218.1584 mph
52.789
103.3343

52.789
165.3694
Comparison of Methods
We ended up with same result for Wonder Woman doing
it in two different ways. Each way requires some work.
You can only use the laws of sines & cosines if:
• you’re dealing with exactly 3 vectors. (If you’re
adding three vectors, the resultant makes 4, and this
method won’t work
• the vectors form a triangle.
Regardless of the method, draw a vector diagram!
Assignment – Non Right Angle Vector
Addition
 Ch. 1 – Pages 24 - 25,
 Problems 26, 29, 31, 32, 45, 46, 51, 52.