Orthogonal vector condition

But here you are dealing with specific vectors, namely ##u+v## and ##u-x##.For example, consider the following equation:$$x+1 = 2x+1$$This equation is true for ##x=0##, but that does not mean that ##x=2x##.In the same way, the equation ##u ⋅ (u + v) = x ⋅ (u + v)## does not necessarily mean that ##u=x## - it is only true if ##u+v=0##. So, you can only conclude that ##u=x## if you have shown that ##u+v=0##. And since you are trying to prove that using the equation
  • #1
terryds
392
13

Homework Statement



Vector u, v, and x are not zero. Vector u + v will be perpendicular (orthogonal) to u-x if

A. |u+v| = |u-v|
B. |v| = |x|
C. u ⋅ u = v ⋅ v, v = -x
D. u ⋅ u = v ⋅ v, v = x
E. u ⋅ u = v ⋅ v

Homework Equations


u⋅v = |u||v| cos θ

The Attempt at a Solution


[/B]
Two vectors are orthogonal to each other if the dot product is zero.

(u + v) ⋅ (u - x) = 0
(u ⋅ u) - (u ⋅ x) + (v ⋅ u) - (v ⋅ x) = 0
u ⋅ (u + v) - x ⋅ (u + v) = 0
u ⋅ (u + v) = x ⋅ (u + v)

u = x

or

u ⋅ (u - x) + v ⋅ ( u - x) = 0
u ⋅ (u - x) = - v (u - x)

so, u = -v

x = -v
v = -x

It seems the answer is C
But, how to get the condition u⋅u = v⋅v
Thanks in advance
 
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  • #2
terryds said:
It seems the answer is C
I don't think so.
Just do it by inspection on the equation (u ⋅ u) - (u ⋅ x) + (v ⋅ u) - (v ⋅ x) = 0. Try the answer choices one by one to find which one satisfies that equation.
 
  • #3
blue_leaf77 said:
I don't think so.
Just do it by inspection on the equation (u ⋅ u) - (u ⋅ x) + (v ⋅ u) - (v ⋅ x) = 0. Try the answer choices one by one to find which one satisfies that equation.

Okay.. Just by inspection.. I get D as the answer..
I thought too far and too much hahahaha...
Thank you :smile:
 
  • #4
blue_leaf77 said:
I don't think so.
Just do it by inspection on the equation (u ⋅ u) - (u ⋅ x) + (v ⋅ u) - (v ⋅ x) = 0. Try the answer choices one by one to find which one satisfies that equation.

Anyway, can you show me the error in my calculation before?
I'm curious where I get wrong
 
  • #5
In both
terryds said:
u ⋅ (u + v) = x ⋅ (u + v)
and
terryds said:
u ⋅ (u - x) = - v (u - x)
you are concluding that if ##(a,b) = (c,b)## for some vector(s) ##b##, then ##a=c## - this conclusion is incorrect. It would have been true had the vector ##b## stands for any vectors in the space.
 
  • Like
Likes terryds

1. What is the definition of orthogonal vector condition?

The orthogonal vector condition, also known as the perpendicular vector condition, states that two vectors are orthogonal (perpendicular) to each other if their dot product is equal to zero.

2. How do you determine if two vectors satisfy the orthogonal vector condition?

To determine if two vectors satisfy the orthogonal vector condition, you can calculate their dot product. If the dot product is equal to zero, then the vectors are orthogonal. Otherwise, they are not orthogonal.

3. What is the importance of the orthogonal vector condition in mathematics?

The orthogonal vector condition is important in mathematics because it allows us to determine if two vectors are perpendicular to each other. This is useful in various applications, such as finding the angle between two vectors or solving systems of equations.

4. Can more than two vectors satisfy the orthogonal vector condition?

Yes, more than two vectors can satisfy the orthogonal vector condition. In fact, any set of vectors that are pairwise orthogonal (i.e. each vector is orthogonal to every other vector in the set) will satisfy the orthogonal vector condition.

5. Is the orthogonal vector condition the same as the perpendicularity condition?

Yes, the orthogonal vector condition and the perpendicularity condition are referring to the same concept. Both conditions state that two vectors are perpendicular if their dot product is equal to zero.

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