Proving a vector inequality in R^n

• king vitamin
In summary, the homework statement is trying to find equations that show that: -x+y <= x+y+2x-x-y <= x-y+2x
king vitamin
Gold Member

Homework Statement

In R^n show that:

$$\Vert\overrightarrow{x - y}\Vert \Vert\overrightarrow{x+y}\Vert \leq \Vert\overrightarrow{x}\Vert^{2} + \Vert\overrightarrow{y}\Vert^{2}$$

Homework Equations

My main attempts have either been from the Triangle Inequality:
$$\Vert\overrightarrow{x+y}\Vert \leq \Vert\overrightarrow{x}\Vert + \Vert\overrightarrow{y}\Vert$$

Or from attempts to implement the idea:
sqrt(a^2 + b^2) <= |a| + |b|

The Attempt at a Solution

Everytime I attempt to do this algebraically, the product on the left becomes a distributive catastrophe and I can't get anything sensible out of it, and attempting to square both sides to get rid of the square root just results in the same catastrophe on the RHS. I tried to represent the left hand side as a dot product but that involved cos(theta) which just complicates the problem. If anyone could point me in the right direction it would be appreciated.

The triangle equality is all you need. Just square both sides and you're almost there already.

I feel more stuck than before! Squaring the Triangle inequality obtains:

$$\Vert\overrightarrow{x+y}\Vert^{2} \leq \Vert\overrightarrow{x}\Vert^{2} + \Vert\overrightarrow{y}\Vert^{2} + 2\Vert\overrightarrow{x}\Vert\Vert\overrightarrow{y}\Vert$$

So do I now need to show that:

$$\Vert\overrightarrow{x - y}\Vert \Vert\overrightarrow{x+y}\Vert \leq \Vert\overrightarrow{x+y}\Vert^{2} - 2\Vert\overrightarrow{x}\Vert\Vert\overrightarrow{y}\Vert$$
? EDIT: I found a counterexample to this equation, so mark it off the list

Even more frustrating is this easy result from the Triangle Inequality:

$$\Vert\overrightarrow{x+y}\Vert\Vert\overrightarrow{x-y}\Vert \leq \Vert\overrightarrow{x}\Vert^{2} + \Vert\overrightarrow{y}\Vert^{2} + 2\Vert\overrightarrow{x}\Vert\Vert\overrightarrow{y}\Vert$$

I just can't get rid of the 2xy term!

Last edited:
Oohhh, now I see, there is a minus sign in one of them.

I did get it to work (more or less) through the dot product. You have to think about cos(theta) a bit, but eventually you will be able to use that a - b <= a if b >= 0 and sqrt(a^2 + b^2) <= |a| + |b| to get where you want.
I hope that's a better hint and you'll get there now.

1. How do you prove a vector inequality in R^n?

To prove a vector inequality in R^n, you need to use mathematical techniques such as the triangle inequality or Cauchy-Schwarz inequality. These techniques involve manipulating the vector components and using algebra to show that one vector is always greater than the other.

2. What is the significance of proving vector inequalities in R^n?

Proving vector inequalities in R^n is important in many areas of science, particularly in physics and engineering. These inequalities can help determine the limits and boundaries of physical systems, and can also be used to optimize solutions in engineering problems.

3. Can you provide an example of a vector inequality in R^n?

One example of a vector inequality in R^n is the Cauchy-Schwarz inequality, which states that for two vectors a and b in R^n, the absolute value of their dot product is always less than or equal to the product of their magnitudes. In other words, |a·b| ≤ |a||b|.

4. What are some common challenges when proving vector inequalities in R^n?

One common challenge when proving vector inequalities in R^n is dealing with higher dimensions. As the number of dimensions increases, the complexity of the problem also increases, making it more difficult to manipulate the vector components and find a solution. Another challenge is identifying which inequality technique to use for a given problem.

5. What are some tips for effectively proving vector inequalities in R^n?

To effectively prove vector inequalities in R^n, it is important to have a strong understanding of vector operations and mathematical techniques such as the triangle inequality and Cauchy-Schwarz inequality. It is also helpful to break down the problem into smaller steps and use visual aids, such as diagrams, to better understand the relationship between the vectors. Additionally, practice and patience are key in mastering this skill.

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