How can I use the dot product formula to find the angle between two vectors?

  • Thread starter leitz
  • Start date
In summary, the conversation discusses a homework problem that requires proving the inequality ||V - U|| <= | ||V||-||U|| | for vectors V and U. The suggested approach is using the dot product formula and simplifying the equation. The expert summarizer provides a hint to use the fact that ||u|| = ||(u - v) + v|| to continue the proof.
  • #1
leitz
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Homework Statement


Prove : ||V - U|| => | ||V||-||U|| |
V and U are vectors

Homework Equations


Maybe the dot product formula: ||A||*||B||cosθ

The Attempt at a Solution


==> ||V - U||2 >= (||V||-||U||)2
==> (V-U) . (V-U) >= ||V||2 +||U||2 - 2||U||*||V||
==> V.V + U.U -2U.V >= V.V + U.U - 2||U||*||V||
==> -2U . V >= -2 ||U||*||V||

I don't know how to go on from here. I always get a nonsensical answer if I try to simplify this. Am I using the right approach?
 
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  • #2
leitz said:

Homework Statement


Prove : ||V - U|| <= | ||V||-||U|| |
V and U are vectors

Homework Equations


Maybe the dot product formula: ||A||*||B||cosθ

You have the inequality backwards above.

Hint: ||u|| = ||(u - v) + v|| ≤ ...
 
  • #3
Can you give me another hint please?
 
  • #4
LCKurtz said:
You have the inequality backwards above.

Hint: ||u|| = ||(u - v) + v|| ≤ ...

leitz said:
Can you give me another hint please?

What did you try with my hint? What might go after the ≤ ?
 

FAQ: How can I use the dot product formula to find the angle between two vectors?

What does the equation ||v -u|| >= | ||v||-||u|| | mean?

The equation ||v -u|| >= | ||v||-||u|| | is a mathematical expression that represents the distance between two vectors v and u. It states that the distance between v and u is greater than or equal to the absolute value of the difference between the magnitudes of v and u.

Why is this equation important in science?

This equation is important in science because it helps us understand the relationship between two vectors. It is often used in physics and engineering to calculate the distance between two points in space, as well as in other fields such as computer science and statistics.

How is this equation used in real-world applications?

The equation ||v -u|| >= | ||v||-||u|| | can be used in various real-world applications such as calculating the speed and direction of moving objects, determining the accuracy of measurements, and analyzing data in statistical models.

What is the significance of the absolute value in this equation?

The absolute value in this equation is important because it gives us the distance between the magnitudes of v and u, regardless of their signs. This allows us to compare the magnitudes of v and u without having to consider their directions.

Can this equation be applied to more than two vectors?

Yes, this equation can be extended to more than two vectors. The general form is ||v1 + v2 + ... + vn|| >= | ||v1|| + ||v2|| + ... + ||vn|| |, where v1, v2, ..., vn are n vectors. This equation is known as the triangle inequality and is applicable in various mathematical and scientific contexts.

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