Find the geometric relation between vectors A and B

In summary, the conversation was about finding the geometric relation between two vectors A and B, given that their magnitudes are equal when added and subtracted. The solution involves using unit-vector notation and the dot product, where the square of the magnitude of a vector is the dot product with itself. By squaring both sides of the given equation and using the distributive property, it can be shown that the dot product of A and B must be equal to 0, meaning that they are perpendicular. This can also be visualized as a parallelogram with diagonals of equal length, indicating a rectangular shape.
  • #1
pf09
5
0
Vectors A and B each lie in the x-y plane. The magnitude of A + B equals the magnitude of A - B. Find the geometric relation between vectors A and B.
(Hint: Express the vectors in unit-vector notation. Use your knowledge of dot products.)

I really don't even know what this problem is asking. What is meant by the geometric relation? I'm not sure how dot products relate. What is my final answer going to be in terms of? Is it going to be an equation? A sentence? I'm really lost. Can someone break down what the objective of this problem is or how to start it?
 
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  • #2
Welcome to PF!

pf09 said:
I'm not sure how dot products relate.

Hi pf09! Welcome to PF! :smile:

The square of the magnitude of a vector is its dot-product with itself: |A|2 = A.A :smile:
I really don't even know what this problem is asking. What is meant by the geometric relation? What is my final answer going to be in terms of? Is it going to be an equation? A sentence? I'm really lost. Can someone break down what the objective of this problem is or how to start it?

Stop philosophising! :rolleyes:

If you actually solve the equation, you should see what the point is. :wink:

So what equation do you get? :smile:
 
  • #3


tiny-tim said:
Hi pf09! Welcome to PF! :smile:

The square of the magnitude of a vector is its dot-product with itself: |A|2 = A.A :smile:Stop philosophising! :rolleyes:

If you actually solve the equation, you should see what the point is. :wink:

So what equation do you get? :smile:
hi, thank you for the welcome.

i know that about the dot product; i just don't know how it relates.
solve which equation? what am i solving for?
i'm not philosophizing. i just don't know what I'm supposed to do.
 
  • #4
i can add vectors, find magnitudes, do dot products, find the angle between vectors,.. i can do all that. i just don't know what this problem is asking. if someone could tell me what I'm supposed to do, i could probably do it. please, i actually dreamt about this problem between my first and last post.
 
  • #5
The "magnitude" of A is the square root of [itex]A\cdot A[/itex] so if the magnitudes of A+ B and A- B are the same you know that [itex](A+B)\cdot(A+B)= (A-B)\cdot(A-B)[/itex] multiply that out and see what it tells you.

One way to do this problem purely geometrically would be to realize, from the "parallelogram" definition of the sum of two vectors, that A+ B is one diagonal of the parallelogram having sides A and B, while A- B is the other diagonal. If those have the same length, what does that tell you about the parallelogram?
 
  • #6
The equation is |A + B| = |A - B|.

Square it, and then use the general forumula |A|2 = A.A
 
  • #7
thank you for these starting points.

ok so after i square both sides of |A + B| = |A - B|, i get what hallsofivy gave me - (A+B).(A+B) = (A-B).(A-B). am i doing this next part right? from there, i said that (A+B).(A+B) = <(a1 + b1)^2, (a2 + b2)^2, (a3 + b3)^2> and (A-B).(A-B) = <(a1 - b1)^2, (a2 - b2)^2, (a3 - b3)^2>. I'm not sure where to go from here.

also, to get back to hallsofivy's help, with a parallelogram having diagonals of the same length, i get a rectangle. so would it be enough for a final answer (after I've shown work) to say that vectors A and B are perpendicular?
 
  • #8
pf09 said:
ok so after i square both sides of |A + B| = |A - B|, i get what hallsofivy gave me - (A+B).(A+B) = (A-B).(A-B). am i doing this next part right? from there, i said that (A+B).(A+B) = <(a1 + b1)^2, (a2 + b2)^2, (a3 + b3)^2> and (A-B).(A-B) = <(a1 - b1)^2, (a2 - b2)^2, (a3 - b3)^2>. I'm not sure where to go from here.

oooh! :cry:

Never use components if you don't have to … that's the joy of vector notation! :wink:

Use the distributive law … (P + Q).R = P.Q + P.R.

Try again! :smile:
 
  • #9
tiny-tim said:
oooh! :cry:

Never use components if you don't have to … that's the joy of vector notation! :wink:

Use the distributive law … (P + Q).R = P.Q + P.R.

Try again! :smile:
haha ok my book still uses components for all the proofs. i have to leave for school now, ill try to get back to this in between classes today. thanks for your help.
 
  • #10
pf09 said:
thank you for these starting points.

ok so after i square both sides of |A + B| = |A - B|, i get what hallsofivy gave me - (A+B).(A+B) = (A-B).(A-B). am i doing this next part right? from there, i said that (A+B).(A+B) = <(a1 + b1)^2, (a2 + b2)^2, (a3 + b3)^2> and (A-B).(A-B) = <(a1 - b1)^2, (a2 - b2)^2, (a3 - b3)^2>. I'm not sure where to go from here.

also, to get back to hallsofivy's help, with a parallelogram having diagonals of the same length, i get a rectangle. so would it be enough for a final answer (after I've shown work) to say that vectors A and B are perpendicular?

Is the dot product of two vectors is a vector or a number?

What is the dot product of two perpendicular vectors?
 

FAQ: Find the geometric relation between vectors A and B

What is the geometric relation between vectors A and B?

The geometric relation between vectors A and B can be described using several different terms, such as parallel, perpendicular, or collinear. It depends on the angle and magnitude between the two vectors.

How do you determine if two vectors are parallel?

To determine if two vectors are parallel, you can use the dot product formula: A • B = ||A|| * ||B|| * cos(theta), where theta is the angle between the two vectors. If the dot product equals 0, the vectors are perpendicular and if it equals ||A|| * ||B||, the vectors are parallel.

Can two vectors be parallel and perpendicular at the same time?

No, two vectors cannot be parallel and perpendicular at the same time. If two vectors are perpendicular, their dot product will be 0. However, if two vectors are parallel, their dot product will not be 0 and they cannot be perpendicular.

What is the difference between collinear and parallel vectors?

Collinear vectors are vectors that lie on the same line, with the same or opposite direction. Parallel vectors, on the other hand, are vectors that have the same direction but may not necessarily lie on the same line.

How does the magnitude of vectors A and B affect their geometric relation?

The magnitude of vectors A and B can affect their geometric relation in terms of being parallel, perpendicular, or collinear. If the magnitude of vector A is larger than vector B, they may not be parallel but they can still be collinear. However, if the magnitude of vector A is equal to vector B, they can be parallel.

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