Adding & Subtracting vectors

[Ammar w]

Homework Statement

If two collinear vectors $\vec{A}$ and $\vec{B}$ are added, the resultant has a magnitude equal to 4.0. If $\vec{B}$ is subtracted from $\vec{A}$, the resultant has a magnitude equal to 8.0. What is the magnitude of $\vec{B}$ ?

Homework Equations

None.

The Attempt at a Solution

|A| + |B| = 4.0 (1)
|A| - |B| = 8.0 (2)
sum the two equations :
2|A| = 12
=> |A| = 6.0
substitute in (1) :
6.0 + |B| = 4.0
=> |B| = -2

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is this a complete and right solution??
should I draw the vectors??

 

[haruspex]

Clearly that's wrong since |B| cannot be negative.

|A| + |B| = 4.0 (1)
|A| - |B| = 8.0 (2)

Let's step back a bit. They're added as vectors:
|A+B| = 4
|A-B| = 8
Since they're collinear, you have equated |A+B| to |A|+|B| etc., but there is another possibility. Can you see what it is?

 

[Ammar w]

Clearly that's wrong since |B| cannot be negative.

Let's step back a bit. They're added as vectors:
|A+B| = 4
|A-B| = 8
Since they're collinear, you have equated |A+B| to |A|+|B| etc.,

Thanks haruspex

so the solution :

|A+B| = 4
|A-B| = 8
because they're collinear :
|A| + |B| = 4
|A| - |B| = 8
sum the two equations :
2|A| = 12
|A| = 6
substitute :
6 + |B| = 4
=> |B| = ?

but there is another possibility. Can you see what it is?

do you mean by drawing??

 

[haruspex]

|A+B| = 4
|A-B| = 8
because they're collinear :
|A| + |B| = 4
|A| - |B| = 8

No, you're still making an assumption that's wrong. What if A and B are in opposite directions?

 

[Nickie]

I would like to point out that there are a few issues with this solution. First, it is important to note that vectors are not just represented by their magnitudes, but also by their direction. So, the equations should be written as |A| + |B| = 4.0 and |A| - |B| = 8.0, where the addition and subtraction take into account the direction of the vectors.

Additionally, the solution provided does not take into account the fact that the vectors are collinear, meaning they have the same direction. Therefore, the magnitude of B cannot be negative, as it would not make sense for a vector to have a negative magnitude.

To solve this problem correctly, we can use the fact that the vectors are collinear to write |A| = k|B|, where k is a constant. Substituting this into the two equations, we get:

k|B| + |B| = 4.0 and k|B| - |B| = 8.0

Solving for k, we get k = 3/4, which means |A| = (3/4)|B|. Substituting this into the first equation, we get:

(3/4)|B| + |B| = 4.0

Solving for |B|, we get |B| = 16/7. Therefore, the magnitude of B is 16/7 units.

To conclude, as a scientist, it is important to pay attention to the details and take into account all the given information when solving a problem. Drawing a diagram can also be helpful in visualizing the problem and arriving at the correct solution.