Begin the slide show.

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…

An ant walks 2.00 m 25° N of E , then turns and walks 4.00 m 20° E of N.

This can’t be solved using our right-triangle math because it isn’t a RIGHT TRIANGLE!

We can add the two individual displacement vectors together by first separating them into pieces, called x- & y-components

The total displacement of the ant…

1) A vector with a -x component and a +y component…

2) A vector with a +x component and a -y component…

3) A vector with a +x component and a +y component…

4) A vector with a -x component and a -y component…

5) A vector with a -x component and a zero y component…

6) A vector with a zero x component and a -y component…

7) For the vector 1350 ft, 30° N of E…

R = 1350 ft

θ = 30°

8) For the vector 14.5 km, 20° W of S…

R = 14.5 km

θ = 70°

9) For the vector 2400 m, S…

R = 2400 m

θ = 90°

An ant walks 2.00 m 25° N of E , then turns and walks 4.00 m 20° E of N.

4.00 m

2.00 m

dt

This was the situation...

The total displacement of the ant…

R1 = 2.00 m, 25° N of E

R2 = 4.00 m, 20° E of N

R1 = 2.00 m, 25° N of E

25°

x = R cosθ = (2.00 m) cos 25° = +1.81262 m

y = R sinθ = (2.00 m) sin 25° = +0.84524 m

1.81262 m

0.84524 m

R2 = 4.00 m, 20° E of N

x = R cosθ = (4.00 m) cos 70° = +1.36808 m

y = R sinθ = (4.00 m) sin 70° = +3.75877 m

1.36808 m

3.75877 m

θ = 70˚

So, you have broken the two individual displacement vectors into components.

Now we can add the x-components together to get a TOTAL X-COMPONENT; adding the y-components together will likewise give a TOTAL Y-COMPONENT.

Let’s review first…

R1 = 2.00 m, 25° N of E

25°

x = R cosθ = (2.00 m) cos 25° = +1.81262 m

y = R sinθ = (2.00 m) sin 25° = +0.84524 m

1.81262 m

0.84524 m

R2 = 4.00 m, 20° E of N

x = R cosθ = (4.00 m) cos 70° = +1.36808 m

y = R sinθ = (4.00 m) sin 70° = +3.75877 m

1.36808 m

3.75877 m

We have the following information:

x y

R1

R2

+1.81262 m

+1.36808 m

+0.84524 m

+3.75877 m

Now we have the following information:

x y

R1

R2

+1.81262 m

+1.36808 m

+0.84524 m

+3.75877 m

Adding the x-components together and the y-components together will produce a TOTAL x- and y-component; these are the components of the resultant.

x y

R1

R2

+1.81262 m

+1.36808 m

+0.84524 m

+3.75877 m

+3.18070 m +4.60401 m

x-component of resultant y-component of resultant

Now that we know the x- and y-components of the resultant (the total displacement of the ant) we can put those components together to create the actual displacement vector.

3.18070 m

4.60401 mdT

θ

The Pythagorean theorem will produce the magnitude of dT:

c2 = a2 + b2

(dT)2 = (3.18070 m)2 + (4.60401 m)2

dT = 5.59587 m 5.60 m

A trig function will produce the angle, θ:

tan θ = (y/x)

θ = tan-1 (4.60401 m / 3.18070 m) = 55º

Of course, ‘55º’ is an ambiguous direction. Since there are 4 axes on the Cartesian coordinate system, there are 8 possible 55º angles.

55º

55º55º

55°

…and there are 4 others (which I won’t bother to show you).

To identify which angle we want, we can use compass directions (N,S,E,W)

3.18070 m

4.60401 mdT

θ

From the diagram we can see that the angle is referenced to the +x axis, which we refer to as EAST.

The vector dT is 55° north of the east line; therefore, the direction of the dT vector would be

55° North of East

So, to summarize

what we just did…

We started with the following vector addition situation…

4.00 m

2.00 m

dt

…which did NOT make a right triangle.

dt

Then we broke each of the individual vectors ( the black ones) into x- and y-components…

…and added them together to get x- and y-components for the total displacement vector.

And now we have a right triangle we can analyze!

Yeah, baby! Let’s give it a try!

Complete #16 on your worksheet. (Check back here for the solution to the problem when you are finished.)

# 16

(continued on next slide)

(west)

(south)

(east)

(north)

(west) (south)

(west) (south)