Requesting help with vector magnitude and direction...?

Question

This was part of a homework assignment for my Physics class. I was wondering if anyone could please help me solve this by providing at least some background guidance:

(Fx is F sub x and Fy is F sub y)

The x and y components of a force are Fx = 12N and Fy = -15N, respectively. Determine the magnitude and direction of F.

Thanks! ^_^

I haven't had much experience in my mathematics background dealing with vectors or anything Physics-related, so I appreciate any advice! =)

Answer 1

F is the vector sum of its components Fx and Fy.

A vector, as you know, has both a magnitude and a direction. It is drawn as a line segment with an arrow head on one end. The end of the line segment with the arrow head is called the "head", and the other end is called the "tail". The direction of the vector is the direction from the tail to the head, and the magnitude is represented as the length of the line segment.

But it doesn't have a position so when you do calculations, or make diagrams, with vectors you can position them anywhere you want. So when you add vectors, you can start with one vector, and position that anywhere you like. It is usually convenient to put the tail of the first vector at the origin (0,0). Then position the tail of the next vector at the head of the first vector, and the sum (or resultant) vector is just the direction and distance from the tail of the first vector to the head of the second one. If you draw a line segment from the tail of the first vector to the head of the second one, and put the arrow head on the end of that line segment where it touches the head of the second vector, then you will have drawn the representation of that vector sum (or resultant vector).

So lets suppose we have two vectors, one of them points along the x-axis and is 10 units long. The other vector points along the y-axis and is also 10 units long. So you can combine these vectors by placing the first one so that the tail is at the origin and the head is at x=10. So its tail is at (0,0) and its head is at (10,0). Then place the second one so that its tail at the head of the first vector. So its tail will be at (10,0) and its head will be at (10,10). Now the sum of those two vectors will be a vector that (as we have positioned it) will have its tail at the origin (0,0) and its head at (10,10).

If you draw that out, you will see that the line segments form a right triangle (and your vector components for your problem will also form a right triangle if you sum them in a similar way). So you can use the Pythagorean theorem to compute the magitude of that vector sum (the length of the line segment that represents the sum), and you can use trigonometry to compute the direction (in this case you know the rise and the run, so you would use the Arctangent function to compute the angle).

In the example I gave, the angle would be 45 degrees (or, in a Physics class that would be called pi/4 radians), and the magnitude would be 10*(square root of 2), or about 14.1 units. Your problem has a different answer, but it is solved in the same way.

Answer 2

Ewry,
You have a right triangle with two sides given and a right angle between them. Your first is the X-direction or 12 to the right (East). The second is 15 to the south (Down). You are looking for the missing side. (12)^2 + (15)^2 = z^2
Thus 144 + 225 = 369 and z^2 = 369. Your side is the square root of 369. The angle will take a little more work. Hopefully you can get help on thuis I must go right now.
Thanks,
Eds


.

Answer 3

If you have some experience with graphing, you can try this. Start your vector at (0,0). Treat the Fx vector as (12,0) and your Fy vector as (0,-15).
Now find (12,-15). The direction of the resultant is from (0,0) to (12,-15) and its magnitude is
sqrt(12^2+15^2)

Answer 4

TO FIND MAGNITUDE:
just square both terms, add them n take sqroot of the addition.
12^2 = 144 and (15)^2 = 225
144 + 225 = 369
sqrt(369) = 19.21 approx

TO FIND DIRECTION:
to find the direction of hypot of a right triangle;
angle = (tan)^-1 (Fy/Fx)
angle = (tan)^-1 (-15/12)
angle = (tan)^-1 (-1.25)
angle = -51.2402