Adding Vectors Using the Component Method

In summary, we discuss the use of component method to add vectors A and B, where the length of vector B is 3.25m and the angle θ = 28.5°. We express the resultant vector A + B in unit-vector notation by adding the x and y components separately. We also explore the idea of adding vectors by breaking them down into perpendicular directions. Overall, this method allows for a simpler and more accurate calculation of resultant vectors.
  • #1
upwardfalling
7
0

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.

p1-38alt.gif


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?
 
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  • #2
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)?
 
  • #3
I don't know if I'm answering your question, but is it 90 degrees..?
 
  • #4
upwardfalling said:
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.
 
  • #5
Cake said:
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?
 
  • #6
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.
 
Last edited:
  • #7
Stephen Hodgson said:
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?
 
  • #8
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)

Stephen Hodgson said:
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 seperately
 
  • #9
Stephen Hodgson said:
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 seperately

Alright, I think I get it! Thank you so much. Hopefully I'll get the right answer
 
  • #10
Cool!

If you post your final answer, I'll happily confirm if it's correct or not
 
  • #11
I got 2.64i + 4.68j , is that correct? I added (Ax + 0, Ay + By) for that answer and wrote it in unit vectors.
 
  • #12
Yep, you've got it :smile:
 
  • #13
Awesome, thanks for the help! :)
 

1. What is a vector?

A vector is a mathematical quantity that has both magnitude (size) and direction. It is represented by an arrow pointing in the direction of the vector, with the length of the arrow representing the magnitude of the vector.

2. What are component vectors?

Component vectors are the individual vectors that make up a larger vector. They are usually represented as horizontal and vertical components, which when added together, form the original vector.

3. How do you find the magnitude of a vector?

The magnitude of a vector can be found using the Pythagorean theorem. This involves squaring the horizontal and vertical components of the vector, adding them together, and then taking the square root of the sum.

4. What is the difference between a scalar and a vector?

A scalar is a quantity that only has magnitude, while a vector has both magnitude and direction. Examples of scalars include temperature and mass, while examples of vectors include velocity and force.

5. How do you add or subtract vectors?

To add or subtract vectors, you can use the parallelogram method or the head-to-tail method. Both methods involve placing the vectors tip-to-tail and then drawing a diagonal from the starting point to the end point of the last vector. The resulting vector is the sum or difference of the original vectors.

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