Sections 1.7 and 1.8

Non Right Triangle Vector Addition

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.)

An ant walks 2.00 m 25° N of E , then turns and walks 4.00 m

20° E of N.

…can not be found using right-triangle math because WE DON’T

HAVE A RIGHT TRIANGLE!

4.00 m

2.00 m

dt

CONSIDER THE FOLLOWING...

The total displacement of the ant…

We can add the two individual

displacement vectors together

by first separating them into

pieces, called x- & y-components

COMPONENTS

Into WHAT?????????

Every vector can be thought of as pointing

somewhat horizontally….

…and somewhat vertically.

They’re kind of like the vector’s shadows.

[This is the black vector’s shadow on the x-axis]

[This is the black

vector’s shadow on

the y-axis]

If we add the x- and y-components

together…

…and it makes a right triangle!

they create the original vector…

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?

8

Murphy's Law:

Anything that can possibly go wrong, will go wrong (at the

worst possible moment).

Cole's Law ??

Finely chopped cabbage

Law of Sines

A

B C

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

a

b c

sin A sin B sin C –––––– = –––––– = –––––– a b 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

10

Now use it to solve the triangle below.

Label sides

and angles

Side c first 15

12.5 =26°

A B

C

c

2 2 2

2 2

2 cos

15 12.5 2 15 12.5 cos 26

c b a a b C

c

Applying the Cosine Law

11

C = 6.65

Now calculate the angles using the Law of Sines.

15 12.5

26°

A B

C

c = 6.65

o2226cos5.12155.1215 c

sin A sin B sin 260 –––––– = –––––– = –––––– 12.5 15 6.65

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)

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?

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.

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.

10

32

vWG

vWA + vAG = vWG 80

vWG

80

32

100

v

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 sin

v 195 =

So, sin = 195 sin 132 / 458, and 18.45

80

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.

Wonder Woman Problem: Component Method

32

10

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

165.3694

52.789

299.3816

103.3

343

165.3694

52.789

103.3343

52.789 165.3694

218.1584 mph

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.

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.