# Vector Math

1. Nov 12, 2013

### Medgirl314

1. The problem statement, all variables and given/known data
A car travels 8 km E, then 8 km N, thrn 16 km W. What is the car's displacement?

2. Relevant equations

a^2+b^2=c^2

3. The attempt at a solution

R^2=8 ^2+8^2
R=11.3 km.

The resultant, R, seems to be 11.3, but I don't think this is the final answer, because I haven't take the 16 km into account yet. What's the next step?

2. Nov 12, 2013

### SteamKing

Staff Emeritus
Grinding through the math is one thing, understanding it is another.

Have you tried making a sketch of the problem?

3. Nov 12, 2013

### phinds

If you follow SteamKing's advice, you'll find the problem to be totally trivial

4. Nov 12, 2013

### Medgirl314

Agreed, I'm never satisified until I find the answer. I drew a diagram, and I'm attempting to find the answer using the parallelogram method. However, my visualization skills don't seem to be kicking in. I have a diagram that does look like a parallelogram, and it can be divided into two right triangles. Working from either side, it appears that the resultant is 11.3, but that doesn't seem complete.

5. Nov 12, 2013

Basically you have a right angled triangle with a and b both equaling 8 units

http://cdn.imghack.se/medium/200a3d66481a86f4584664ee2dbd0160.jpg

6. Nov 12, 2013

### Medgirl314

Great, thank you! I suppose the 16 came in indirectly. I thought it was odd because the answer is the same as one of my practice problems, so it just seemed too easy.

7. Nov 12, 2013

### Medgirl314

So the distance traveled would be 32 km and if the trip took three hours, the average veocity would be 10.6 km/h NE, correct? And the speed would be 10.6 km/h?

8. Nov 12, 2013

### haruspex

There's an interesting distinction between average speed and average velocity when the path is not a straight line. For average speed you take the total distance travelled along the path, but for average velocity you just look at the vector connecting start and end points - the path is irrelevant.

9. Nov 12, 2013

### Medgirl314

Are you saying that my answer is correct, but the NE part isn't needed? My physics teacher noted that it was important, though that may go against the norm.

10. Nov 12, 2013

### cepheid

Staff Emeritus
No, what haruspex was saying was that the answer for the average speed and the average velocity are not the same. For speed, you take the distance travelled divided by time, but for velocity, you use the displacement, a vector, which is the difference between the final and initial position vectors, and divide this by time.

11. Nov 12, 2013

### Medgirl314

Thank you! Would the velocity be 11.3 km/3 hours?

12. Nov 12, 2013

### cepheid

Staff Emeritus
Yes, but since displacement is a vector, you need the direction as well.

13. Nov 12, 2013

### Medgirl314

Right, so would that be ENE? Judging by the direction of my resultant?

14. Nov 12, 2013

The displacement vector is going north west. (Look at the img I attached above - the hypotenuse)

It may be better to use ° though, depends on preference.

15. Nov 12, 2013

### cepheid

Staff Emeritus
Look at FaraDazed's drawing. On that page, straight up is north, straight left is west, etc. What direction is your resultant on the page. More importantly, what angle does it make with the horizontal?

16. Nov 12, 2013

### cepheid

Staff Emeritus
FaraDazed, we don't give complete solutions or give away answers on PF. Since you've done so, we have no idea if the OP understands how to determine the direction.

17. Nov 12, 2013

### Medgirl314

Okay, I see it now. I looked at my drawing, which was flawed. It would be NW. Or maybe phrashing it like my teacher does, WNW.

18. Nov 12, 2013

### cepheid

Staff Emeritus
Actually, NW, and WNW are two totally different directions. A precise way to specify the direction is with an angle. Suppose we measure angles clockwise from vertically up (north). Then N means the vector points at 0°, and W means the vector points at -90°. What direction is NW? How about WNW? What direction is your vector pointing in?

19. Nov 12, 2013

### Medgirl314

Oh, I didn't know that! Thank you. My teacher just said "Well, this resultant is pointing slightly more to the right than up, so it's ENE." Which made sense, it just wasn't explained. The degree explanation helps. Mine was pointing NW because my drawng was flawed.

20. Nov 12, 2013

### cepheid

Staff Emeritus
Right, so can you tell me what angle your displacement vector points at, and what compass direction that corresponds to?

Northwest (NW) means "exactly between north and west". So, since N is 0 deg. and W is -90 deg, NW would be halfway between that, which would be -45 deg (or 360 - 45 = 315 deg, if you want to express angles as being between 0 and 360 only). This is NW.

West-northwest means "halfway between west and northwest". So, since west is -90 deg. and NW is -45 deg, halfway between that would be 22.5 degrees from each one. -45 - 22.5 deg = -67.5 deg.
OR, you could express the angle as 360 - 67.5 = 292.5. This is WNW.