Adding Vectors Using the Component Method

[upwardfalling]

Homework Statement

Use the component method to add the vectors vector A and vector B shown in the figure. The length of vector B is 3.25 m and the angle θ = 28.5°. Express the resultant vector A + vector B in unit-vector notation.

Homework Equations

x = rcos
y = rsin

The Attempt at a Solution

I drew the components for vector A, and got Ax = 3cos28.5 = 2.64m and Ay = 3sin28.5 = 1.43m ( i don't know if they are correct though). I'm lost at what to do next for vector B cause its a vertical line.

Is it right for me to say that after finding vector B's components, i add the X components to find i unit vector and add the Y components to find the j unit vector? And i just get the answer by putting them in one equation, i + j?

 

[Cake]

Well, if it's perpendicular to the x-axis think of what is the angle for it's measure (if the positive x-axis is 0)?

 

[upwardfalling]

I don't know if I'm answering your question, but is it 90 degrees..?

 

[Cake]

I don't know if I'm answering your question, but is it 90 degrees..?

Yes. Good. So now you can calculate the components of the B vector. It's ok if one of them is zero.

 

[upwardfalling]

Yes. Good. So now you can calculate the components of the B vector. It's ok if one of them is zero.

I'm not so sure about how to calculate the components. As in isn't there not enough information?

I don't know if i can just assume Bx = 0 and By = 3.25m, but if i can, will it be Ax + Bx = 2.64 + 0 = 2.64i and Ay + By = 1.43 + 3.25 = 4.68j?

 

[Matternot]

It is 90 degrees but think about what x=rcosθ and y=rsinθ actually means; don't just plug into the formulae and hope.

The vector is being split up into 2 vectors in perpendicular directions: (x,y). Here you calculated A as (2.64m,1.43m). This means, if I move 2.64 in the x direction and 1.43 in the y direction, you will end up where the vector points (i.e. 3m away at 28.5 degrees).

If my vector was (a,b) and I wanted to add (c,d) to it, this means going a in the x direction, then b in the x direction, then c in the x direction, then d in the y direction. This is clearly a+c in the x direction and b+d in the y direction. Therefore this results in (a,b)+(c,d)=(a+c,b+d).

If B is a vertical line, it only has a y component. You don't need to travel in the x direction at all to reach the end of the arrow. This means the vector will be (0,3.25). i.e. 0 in the x direction, 3.25 in the y direction. It is easier to think about this than the angle and magnitude of b.

You are therefore doing the addition A+B=(rcosθ,rsinθ)+(0,By), which I shall leave to you to calculate.

I hope this helps.

 

[upwardfalling]

It is 90 degrees but think about what x=rcosθ and y=rsinθ actually means; don't just plug into the formulae and hope.

The vector is being split up into 2 vectors in perpendicular directions: (x,y). Here you calculated A as (2.64m,1.43m). This means, if I move 2.64 in the x direction and 1.43 in the y direction, you will end up where the vector points (i.e. 3m away at 28.5 degrees).

If my vector was (a,b) and I wanted to add (c,d) to it, this means going a in the x direction, then b in the x direction, then c in the x direction, then d in the y direction. This is clearly a+c in the x direction and b+d in the y direction. Therefore this results in (a,b)+(c,d)=(a+c,b+d).

If B is a vertical line, it only has a y component. You don't need to travel in the x direction at all to reach the end of the arrow. This means the vector will be (0,3.25). i.e. 0 in the x direction, 3.25 in the y direction. It is easier to think about this than the angle and magnitude of b.

You are therefore doing the addition A+B=(rcosθ,rsinθ)+(0,By), which I shall leave to you to calculate.

I hope this helps.

I think I understand but am a little confused at the same time. It did help me in understanding why By = 0 though! And by A+B=(rcosθ,rsinθ)+(0,By), you do mean that I add the x components and y components separately, am I correct?

 

[Matternot]

Exactly. They can be added separately.

This is because if my vector was (a,b) and I wanted to go one more step in the x direction, the y value wouldn't be effected at all. my new vector would be (a+1,b)

You can also think of (a,b) as (a,0)+(0,b) if that helps. Then (a,b)+(1,0) is the same as (a,0) + (1,0) + (0,b) = (a+1,0)+(0,b) = (a+1,b)

If my vector was (a,b) and I wanted to add (c,d) to it, this means going a in the x direction, then b in the x direction, then c in the x direction, then d in the y direction. This is clearly a+c in the x direction and b+d in the y direction. Therefore this results in (a,b)+(c,d)=(a+c,b+d).

As I said before, (a,b)+(c,d) = (a+c,b+d)
Here the x and y components are being added separately

 

[upwardfalling]

Exactly. They can be added separately.

This is because if my vector was (a,b) and I wanted to go one more step in the x direction, the y value wouldn't be effected at all. my new vector would be (a+1,b)

You can also think of (a,b) as (a,0)+(0,b) if that helps. Then (a,b)+(1,0) is the same as (a,0) + (1,0) + (0,b) = (a+1,0)+(0,b) = (a+1,b)As I said before, (a,b)+(c,d) = (a+c,b+d)
Here the x and y components are being added separately

Alright, I think I get it! Thank you so much. Hopefully I'll get the right answer

 

[Matternot]

Cool!

If you post your final answer, I'll happily confirm if it's correct or not

 

[upwardfalling]

I got 2.64i + 4.68j , is that correct? I added (Ax + 0, Ay + By) for that answer and wrote it in unit vectors.

 

[Matternot]

Yep, you've got it

 

[upwardfalling]

Awesome, thanks for the help! :)