Did I do this right? What does the angle 42 degrees have to do with this problem?

Question

Vector A is 34 Newtons NW, vector B is 20 Newtons at an azimuth of 42 degrees (azimuth is measured clockwise from north)

What is the resultant vector for A + B?

34 Newtons + 20 Newtons = 54 Newtons

What is the equilibrant vector for A - B

34 Newtons - 20 Newtons = 14 Newtons

Answer 1

So with this problem, vectors have two things: magnitude and direction

An example of helping you understand is talking about distance and displacement. Distance is just adding the magnitude and displacement is talking about vectors.

Ex: You walk north for 4 miles to buy groceries, then you walk south for 4 miles to go back to your house. The Distance is just magnitude so 4+4 = 8 miles. However, the displacement is about vectors so you have a vector going up with magnitude of 4 and a vector going down with a magnitude of 4 so thinking about the direction, it'll be 4-4 = 0. That's why the direction is important with vectors. Another way of thinking about displacement is where you start and where you end and finding the distance between the two. So if you start at home and end at home, the displacement is 0 although your legs feel like they walked a total of 8 miles which is distance.

Now for this problem, you'd need to separate the vectors into x-y components using trigonometry:

A = 34N NW [which means 45 degrees from the azimuth/the x-axis]

B = 20N 42 degrees from azimuth [which is 48 degrees from the x-axis]

[I would suggest the first thing to do is to draw it out]

Let's start with the y component:

Ay = 34 sin(45)

By = 20 sin(48)

Since both directions point up, you can just add them: Ay+By = Cy = 34 sin(45) + 20 sin(48)

Let's go onto the x component:

Ax = 34 cos(45) [to the left]

Bx = 20 cos(48) [to the right]

Since they are opposite directions, we usually take the left to be negative so:

-Ax + Bx = Cx = -34 cos(45) + 20 cos(48)

Now we have the x and y components of C (called the resultant vector)

and if you have the x and y of a line, it make a right triangle with x and y being the sides so to find the hypotenuse, just use the Pythagorean theorem:

Cx^2+Cy^2 = C^2

C = Sqrt(Cx^2+Cy^2)

And that's the magnitude of the C

[And make sure to include the direction of C as well which can be found using tan(Cy/Cx)]

Now for A-B, the - just flips the direction of B so instead of going up and to the right, it's going down and to the left [keeping the same angle to the azimuth]. To solve, just do the same steps as stated above and adjust for the direction change.