Math problem! vectors?

Question

Resultant force
Three forces with magnitudes of 75 pounds, 100 pounds, and 125 pounds act on an object at angles of 30, 45, 120, respectively, with the positive x-axis. Find the direction and magnitude of the resultant of these forces.

Answer 1

It helps to start with a sketch when doing these physics applications. See here: http://img529.imageshack.us/img529/295/diagram1ls7.jpg

There are a couple different ways to do this, but the most straightforward way is to get the three given vectors into component form and add them. Treat them like three different triangles. The magnitude of each force will be the length of the hypotenuse and the components will be the legs of the triangle. The y component for each is magnitude(sine of angle) and the x component for each is magnitude (cosine of angle). Make sure to pay attention to where the arrows are pointing. One is pointing into the second quadrant, meaning it has a NEGATIVE x component and a positive y component.
Here are sketches of the three triangles I'm talking about:
75 pound force: http://img454.imageshack.us/img454/3994/diagram2xk7.jpg
100 pound force: http://img213.imageshack.us/img213/9760/diagram3yo8.jpg
125 pound force: http://img505.imageshack.us/img505/20/diagram4qg7.jpg

See that each force has components?
Go ahead and calculate them:
75 pound force:
x component = 75cos(30) = 64.95
y component = 75sin(30) = 37.5
100 pound force:
x component = 100cos(45)= 70.71
y component = 100sin(45)= 70.71
125 pound force:
x component = -125cos(60)= -62.5
y component = 125sin(60)= 108.25

To find the magnitude of the resultant, first add the x and y components of the vectors to get the x and y components of the resultant.
x(resultant) = 64.95 + 70.71 - 62.5 = 73.16
y(resultant) = 37.5 + 70.71 + 108.25 = 216.46

You can make a resultant triangle just like the ones for the other forces. Again, the x and y components are the legs, and the resultant's magnitude is the hypotenuse:
http://img294.imageshack.us/img294/9256/diagram5uc8.jpg
Just use the pythagorean theorem to find the hypotenuse of the triangle and consequently the magnitude of the resultant:
R^2 = x^2 + y^2
R^2 = 73.16^2 + 216.46^2
R = squareroot[73.16^2 + 216.46^2]
R = 228.5 lb.

Now all that's left is the direction.
Basically this is just the angle at which the resultant force is acting (In the diagram I called it the "angle of action"). It's just the inverse tangent of the legs.
tan(angle) = (y/x)
tan(angle of action) = (216.46/73.16)
angle of action = arctan(216.46/73.16)
angle = 71.34 degrees
So the direction is 71.34 degrees with respect to the [positive] x axis.
hope this helps!

Answer 2

a million. |a| = a million, |b| = ?3, a,b perpendicular i) |( a million??? + b) x (a - b)| ought to be some vector ii) homes of the pass multiplication: a million. a x a = 0 (vector) 2. a x b = – b x a (2a + b)x(a - 2b) = 2 a x a – 4 a x b + b x a – 2b x b = 4 b x a + b x a = 5 b x a |5 b x a| = 5 |b| · |a| · sin(90deg) = 5·a million·?3·a million = 5?3. 2) (I - X)^(-a million) = I + X + X² Multiply the two facets via (I - X) from the left: (I - X) * (I - X)^(-a million) = (I - X) * (I + X + X²) I = I + X + X² - X*I - X*X - X*X² I = I + X + X² - X - X² - X³ I = I - X³ on condition that X³ = 0 ===> we acquire I = I id. you may verify that (I - X)^(-a million) = I + X + X² additionally supplies id while multiplying the two facets via (I - X) from the main appropriate.

Answer 3

75 pounds force (30°)
x component = 65 lbs
y component = 37.5 lbs

100 pounds force (45°)
x component = 70.7 lbs
y component = 70.7 lbs

125 lbs force (120°)
x component = - 62.5 lbs
y component = 108.2 lbs
RESULTANT
x component = 73.2 lbs
y component = 216.4 lbs
magnitude = √ [(73.2)² + (216.4)² ]
magnitude = 228.4 pounds
angle = tan^(-1)(216.4 / 73.2) = 71.3°