Finding the resultant of two vectors?

Question

Find the resultant of these two vectors: 2.00 x 10^2 units due east and 4.00 x 10^2 units and 30.0 degrees north of west

I'm in a coorespondance course that does a horrible job explaining how to do this, so if anyone could help me out with this it would be greatly appreciated.

Answer 1

Trigonometry.
Simply draw the two vectors tip to tail, then the resultant from the beginning of the first to the end of the last.

You can determine the magnitude using the law of cosines for your resultant vector.

c^2 = a^2 + b^2 -2abCos(C)

c = sqrt(200^2 + 400^2 - 2(200)(400)Cos(30))

c= sqrt(40000 + 160000 - 160000(0.866025404))

c = sqrt(200000-138564) = sqrt(61436)

c = 247.86 or 2.48 x 10^2 units

For the angle, you would use the Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C)

248/sin(30) = 400/sin(A)

A = arcsin((400)sin(30)/248) = arcsin(0.80645)

A = 53.75deg and 126.25deg (180-A)

since we are looking for and angle of an obtuse triangle, the answer is the second value

R = 2.48 x 10^2 units @ 126.25deg

Answer 2

You should apply here the law of parallelogram, which states that:

R^2 = P^2 + Q^2 + 2PQcos a

where P= 2.00 x 10^2

Q= 4.00 x 10^2

a= angle between P and Q
here it is 150 degre
( As You are describing)

R= Resultant required. (in magnitude)

b = Angle between P and R

Substitute the values in above formulae. Which gives magnitude of R.

Now it requires to find out the direction also.

So, tan b = Q sin a/ ( P+ Qcos a)

Substitute the values and then find inverse of tan, which would give angle in degrees.

Good Luck.

Answer 3

First, note that {flick} made a mistake in his diagram. You want the _sum_ of the vectors; but {flick}'s diagram shows the _difference_ of the vectors.

Sometimes the easiest way to add vectors is to express each vector in terms of its x- and y- components (or, in this case, its east and north components), and then add those separately. In your example:

Vector 1:
east component: 200 units
north component: 0 units

Vector 2:
east component: -400(cos(30)) units [negative because it's actually leaning toward the west]
north component: 400(sin(30)) units

You will need to brush up on the definition of "sine" and "cosine" if you don't already know what they are--they are simple but indispensible for separating vectors into their components.

Having done the separation, you can add the north components and east components separately:

east component of resultant:
200 units + -400(cos(30)) units = -146.4 units

north component of resultant:
0 units + 400(sin(30)) units = 200 units

That's technically a complete description of the resultant. But if they want the answer expressed in terms of the resultant's length and angle, you can do it this way:

By the pythagorean theorem:
length = sqrt((-146.4)² + 200²) = 247.9 units

tan(angle) = 200/146.4 = 1.366
From this, the angle is 53.8 degrees north of west.

Answer 4

Well, the easiest method to get this is to plot its graph.

It will look something like this:
http://img411.imageshack.us/img411/3031/untitledea7.gif

Draw this on a graph paper. Taking actual scale. Then measure the blue line.

PS: Take 4 x 10^2 as 8 cm and 2 x 10^2 as 4 cm. Then multiply the length of the blue line by 5 x 10^1 to get it in the form of x.xx * 10^2.

Hope it helps