Facing problem in a questions on Vectors

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In summary, by squaring both sides of the equation and using the dot product, it can be shown that vector a is perpendicular to vector b. This is because the dot product of two perpendicular vectors is 0, and by equating the squared equations of (a+b) and (a-b), the squared terms cancel out and leave only 4(a.b)=0. This shows that the angle between the two vectors is 90 degrees, making them perpendicular. While there may be other ways to solve this problem, using the dot product is a simple and effective solution.
  • #1
ritik.dutta3
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If |vector a+vector b|= |vector a- vector b| show that vector a is perpendicular to vector b.





A friend of mine suggested me to square both the sides of the equation. At the end, the result was vector a. vector b=0, from which it could be proved that the vectors are perpendicular. But why do we use the dot product and not the cross product?
 
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  • #2
How did you use the dot product here? And how would use the cross product?
 
  • #3
voko said:
How did you use the dot product here?
He squared them and expanded in (a[itex]\pm[/itex]b)2 and applied dot product at 2(a.b)
 
  • #4
ritik.dutta3 said:
At the end, the result was vector a.
How come? After squaring, the squared magnitudes get canceled and you get 4abcosθ=0.
 
  • #5
PhysicoRaj said:
He squared them and expanded in (a[itex]\pm[/itex]b)2 and applied dot product at 2(a.b)

Please understand that the questions are intended for the original poster (OP). The purpose of those questions is to let the OP show what the OP knows and whether the OP knows that correctly. Your answering those questions for the OP does not help me or the OP in any way, and is not appreciated. No offence, please.

ritik.dutta3, please answer my original questions.
 
  • #6
but why do we have to use dot product and not cross product?
 
  • #7
The definition of A.B = |A||B|cos(theta) where theta is the angle between the two vectors so if A is perpendicular to B then theta = 90 degrees and the cos(theta) = 0.

In contrast, the definition of AxB = |A||B|sin(theta) so for A is perpendicular to B then sin(theta) = 1 and
AxB = |A||B|

So given A and B, we can create a parallelogram with A and B as sides so that one diagonal is A+B and the other is A-B. Geometrically the diagonals of a parallelogram are always perpendicular and for them to be the same length then the parallelogram must be a square and hence A is perpendicular to B.

I can't see why you couldn't have used the cross product except for the fact that it is further defined to create a vector perpendicular to both and this would confuse the issue a bit.
 
  • #8
voko, here is what i did:

(a+b)2= a2+b2+2(a).(b).

(a-b)2=a2+b2-2(a).(b).

on equating them, the square terms get canceled and what remains is this:
4(a).(b)=0
(a).(b)=0

since in a dot product if the product is equal to 0, the angle between them should be 90 degrees.
but why do we have to use the dot product over here?
 
  • #9
voko said:
Please understand that the questions are intended for the original poster (OP). The purpose of those questions is to let the OP show what the OP knows and whether the OP knows that correctly. Your answering those questions for the OP does not help me or the OP in any way, and is not appreciated. No offence, please.
Oops! sorry..:redface:
 
  • #10
ritik.dutta3 said:
voko, here is what i did:

(a+b)2= a2+b2+2(a).(b).

(a-b)2=a2+b2-2(a).(b).

on equating them, the square terms get canceled and what remains is this:
4(a).(b)=0
(a).(b)=0

since in a dot product if the product is equal to 0, the angle between them should be 90 degrees.
but why do we have to use the dot product over here?

You do not have to use it. But you did, and it worked. Can you think of any other way to solve the problem?
 
  • #11
No voko, i cannot.
 
  • #12
So you have a simple solution involving the dot product and no other solution without the dot product. I think it is reason enough why the dot product was used. I would stop worrying, because there is really nothing to worry about.
 
  • #13
it is a rectangle...
 

Related to Facing problem in a questions on Vectors

1. What are vectors and how are they used?

Vectors are quantities that have both magnitude (size) and direction. They are commonly used in mathematics and physics to represent physical quantities such as velocity, force, and displacement.

2. What are the basic operations of vectors?

The basic operations of vectors include addition, subtraction, and scalar multiplication. Addition of vectors involves adding their respective components, while subtraction involves subtracting the components. Scalar multiplication involves multiplying a vector by a scalar (a real number).

3. How do I determine the magnitude and direction of a vector?

The magnitude of a vector can be determined using the Pythagorean theorem, where the square root of the sum of the squared components gives the magnitude. The direction of a vector can be determined using trigonometric functions, such as tangent or sine.

4. How are vectors represented graphically?

Vectors can be represented graphically using arrows, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction of the vector. The tail of the arrow is usually placed at the origin of a coordinate system.

5. Can vectors be added or subtracted if they have different dimensions?

No, vectors cannot be added or subtracted if they have different dimensions. In order to perform these operations, the vectors must have the same number of components and the same units.

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