Proving Geometric Fact: u+v Perpendicular to u-v Using Dot Product

In summary: Vector addition is distributive over addition of vectors. So, u+ v+ w will be the same as (u+ v)+ (w+ u), etc.In summary, the question is asking for someone to calculate the dot product of two vectors and explain why it is perpendicular if they are perpendicular.
  • #1
Teggles
3
0
"If u and v are any two vectors of the same length, use the dot product to show that
u + v is perpendicular to u − v. What fact from geometry is does this represent."

This is basically the last question in an assignment on vectors (first year university, linear algebra). The questions all focus on things like equations of planes, angles of intersection etc. that require little insight. This one however seems to require quite a lot of insight.

I understand how to calculate the dot product, cross product, length of vectors, angle between vectors, etc., but I don't even know where to start here.

If anyone could help me through this, it'd be extremely appreciated :)
 
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  • #2


Well, I hate to be obtuse but if you "understand how to calculate the dot product" and the problem tells you to use the dot product, isn't it obvious that you should start by taking the dot product?

What is the dot product [itex](u- v)\cdot(u+ v)[[/itex]?
 
  • #3


Well...

(u - v) . (u + v) = |u - v| |u + v| cos θ

Of course, I don't know |u - v| or |u + v|, only that |u| and |v| are the same. I don't know how this bit of information is meant to be used.

If the vectors are perpendicular, θ will be 90 degrees, and so dot product will be 0. Is my aim to therefore prove that the dot product is 0? How? Though the other method of calculating the dot product? I'm sorry, I just feel clueless.
 
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  • #4


Try writing out the vectors' entries, i.e. u = (u1, u2, u3) and v = (v1, v2, v3) and use the entry-wise definition of the dot product. Also try writing out what |u| = |v| means in terms of the entries.
 
  • #5


No! You don't have to be as complicated as either of those. [itex](u+ v)[/itex][itex]\cdot(u- v)[/itex][itex]= u\cdot u[/itex][itex]- u\cdot v[/itex][itex]+ v\cdot u- v\cdot v[/itex][itex]= |u|^2- |v|^2[/itex] because the dot product is commutative so that [itex]-u\cdot v+ v\cdot u= 0[/itex]
 
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  • #6


Awesome. I managed to write a full proof that it would equal 0. I'm fairly glad you omitted some of the reasoning/detail because it actually made me think about and understand each step for myself. Thanks for your help!

Still haven't worked out what "geometric fact" the question wants, but I'll work on it.
 
  • #7


Think about the "parallelogram" rule for adding vectors. If u and v are two sides of a parallelogram, what are u+ v and u- v?
 

1. How do you prove that u+v is perpendicular to u-v using the dot product?

The dot product of two vectors u and v is calculated by multiplying their corresponding components and then summing the results. To prove that u+v is perpendicular to u-v, you need to show that their dot product is equal to 0, which indicates that the two vectors are perpendicular to each other. This can be achieved by expanding the dot product equation and using the property that the dot product of perpendicular vectors is 0.

2. What is the significance of the dot product in proving geometric facts?

The dot product is an important tool in proving geometric facts because it allows us to determine the angle between two vectors and whether they are perpendicular. This is because the dot product is equal to the product of the magnitudes of the two vectors and the cosine of the angle between them. It also helps in calculating projections, distance, and work done by a force in physics and engineering applications.

3. Can the dot product be used to prove other geometric facts?

Yes, the dot product can be used to prove various geometric facts, such as whether two vectors are parallel, finding the angle between two lines, and determining whether a point lies on a line. It is a versatile tool that can be applied to many geometric problems and can be extended to higher dimensions.

4. Are there any other methods to prove the perpendicularity of two vectors?

Yes, there are other methods to prove the perpendicularity of two vectors, such as using the cross product, which results in a vector that is perpendicular to both of the original vectors. Another method is to use the slope of the lines formed by the vectors and showing that they are negative reciprocals of each other. However, the dot product method is often the simplest and most straightforward way to prove perpendicularity.

5. How can this geometric fact be applied in real-world situations?

The fact that u+v is perpendicular to u-v using the dot product has several real-world applications. For example, in physics, this fact is used to calculate the work done by a force on an object by taking the dot product of the force vector and the displacement vector. In engineering, it is used to determine the moment of a force about a point. It is also used in geometric problems, such as finding the shortest distance between a point and a line or determining the angle between two planes.

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