Adding Vectors: Understanding Direction and Length

In summary: If you want to use the angle between the resultant and the horizontal, you can subtract the angle that the original triangle makes with the horizontal from 90°.In summary, when adding two vectors using their components, the order in which the components are placed does not affect the resultant vector. However, the angle of the triangle formed by the components may change. To determine the angle of the resultant vector, one can subtract the angle that the original triangle makes with the horizontal from 90°.
  • #1
aatari
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3

Homework Statement


Hi Guys, I am a bit unclear regarding adding vectors and hoping someone can clear up the confusion for me.

In the image below, we are adding two vectors and we used the vector components to find find x and y. Finally we then used x and y to get the resultant and the angle.

There is a diagram after "Now place the resultant components head-to-tail to form a right angled triangle", where we begin with x going east, then y going north and touching the tip of x.

My question is how do we determine to place the x and y in this way? Is there any criteria or methodology that we use because I could also place x starting at the tip of resultant vector and going east and then from the head of x vector going north to form y vector. If I do this the angle differs as adjacent and opposite sides of the angle change. Could anyone please help me understand this.

I hope my questions is clear.Problem: Determine the length and direction of a + b if a is 4 m [N30°E] and b is 2 m [S40°W].

Homework Equations


N/A

The Attempt at a Solution


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  • #2
aatari said:
Is there any criteria or methodology that we use because I could also place x starting at the tip of resultant vector and going east and then from the head of x vector going north to form y vector.
It doesn't matter whether you place the start of the y component vector at the tip of the x component vector (not the resultant vector!), or the reverse. You get the same answer either way.

aatari said:
If I do this the angle differs as adjacent and opposite sides of the angle change.
Try again. If you get different resultant vectors, you're doing something wrong. (Note that the triangle you get might be different, but the resultant vector will be the same.)
 
  • #3
aatari said:
My question is how do we determine to place the x and y in this way? Is there any criteria or methodology that we use because I could also place x starting at the tip of resultant vector and going east and then from the head of x vector going north to form y vector. If I do this the angle differs as adjacent and opposite sides of the angle change. Could anyone please help me understand this.
I'm not sure I understand your question, but the key here is to combine the x and y vectors. You have a y component that is positive 1.93, towards the North. You have a positive 0.71 x component, towards the East. So whether you put the x arrow down and then place the tail of the y with the head of the x, or you put the y arrow down first and then place the x at its end, you still end up in the same place. That same place is 1.93 to the North and 0.71 to the East of where ever you started from. Also, in either case, once you draw the resultant arrow, you will have a right triangle and you will be able to use their formula to compute the length of that arrow (2).

However, if you use the different triangle, you'll be looking at a different angle. Instead of computing 20 degrees east of north (as shown in the example) you could end up with 70 degrees north of east. But that would just be an alternate way of specifying the same bearing.
 
  • #4
Doc Al said:
It doesn't matter whether you place the start of the y component vector at the tip of the x component vector (not the resultant vector!), or the reverse. You get the same answer either way.Try again. If you get different resultant vectors, you're doing something wrong. (Note that the triangle you get might be different, but the resultant vector will be the same.)

Your are right, resultant does not change. However, my angle changes.
 
  • #5
.Scott said:
However, if you use the different triangle, you'll be looking at a different angle. Instead of computing 20 degrees east of north (as shown in the example) you could end up with 70 degrees north of east. But that would just be an alternate way of specifying the same bearing.

So my answer will still be correct?
 
  • #6
aatari said:
However, my angle changes.
The angle of your triangle changes. But the angle that the resultant makes with the horizontal and the vertical does not. You may have to convert one to the other.
 
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Related to Adding Vectors: Understanding Direction and Length

1. What is a vector?

A vector is a mathematical representation of both magnitude (length) and direction. It is often represented by an arrow, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction.

2. How do you add vectors?

To add vectors, you must first determine the direction and magnitude of each vector. Then, you can add the magnitudes of the vectors together and use trigonometry to find the direction of the resulting vector.

3. Can you add more than two vectors at a time?

Yes, you can add as many vectors as you want at a time. The process is the same as adding two vectors, but you must consider the direction and magnitude of each vector in the calculation.

4. Can vectors be negative?

Yes, vectors can have both positive and negative values. The sign indicates the direction of the vector, with positive values representing a direction to the right or up, and negative values representing a direction to the left or down.

5. Why is understanding direction and length important in adding vectors?

Understanding direction and length is crucial in adding vectors because it allows us to accurately represent and calculate the resulting vector. Without considering direction and length, the resulting vector may be incorrect and not reflect the actual physical motion or force being represented.

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