Understanding the Relationship between Dot and Cross Products

In summary, the conversation discusses finding the value of (u• v) 2 + ||u × v||2 given the magnitudes of u and v. The solution involves squaring both the dot and cross product equations and simplifying to 225. The conversation also touches on the relationship between cosine and sine, which leads to the simplified solution. However, the concept of squaring both the dot and cross product equations is purely mathematical and does not have a real-world application.
  • #1
zr95
25
1

Homework Statement


Find (u• v) 2 + ||u × v||2 , given ||u|| = 5 and ||v|| = 3

Homework Equations


u•v=||u|| ||v|| cos(x)
u×v=||u|| ||v|| sin(x)

The Attempt at a Solution



Using these two equations I squared them both, brought them together and ended up with 225cos2(x) + 225sin2(x) and received a final answer of 225.

Am I correct in my math?

Why is it that if you square both the dot product and the cross product you just get the two magnitudes squared and multiplied together?
 
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  • #2
zr95 said:

Homework Statement


Find (u• v) 2 + ||u × v||2 , given ||u|| = 5 and ||v|| = 3

Homework Equations


u•v=||u|| ||v|| cos(x)
u×v=||u|| ||v|| sin(x)

The Attempt at a Solution



Using these two equations I squared them both, brought them together and ended up with 225cos2(x) + 225sin2(x) and received a final answer of 225.

Am I correct in my math?

Why is it that if you square both the dot product and the cross product you just get the two magnitudes squared and multiplied together?

cos(x)^2 + sin(x)^2 = 1

Remember that. It comes up very frequently.
 
  • #3
Hornbein said:
cos(x)^2 + sin(x)^2 = 1

Remember that. It comes up very frequently.
Yes that's what I used to simplify down to 225.
 
  • #4
zr95 said:
Yes that's what I used to simplify down to 225.

Oh, right. Anyway, I can't come up with a better answer to your question.
 
  • #5
Hornbein said:
Oh, right. Anyway, I can't come up with a better answer to your question.
I guess I was more curious about why this holds true in terms of words as opposed to the mathematics.

(u• v) 2 + ||u × v||2 = ||u||2 * ||v||2
 
  • #6
zr95 said:
I guess I was more curious about why this holds true in terms of words
To do that, you'd have to start with descriptions of dot and cross product that do not depend on the algebra.
 
  • #7
zr95 said:
I guess I was more curious about why this holds true in terms of words as opposed to the mathematics.

(u• v) 2 + ||u × v||2 = ||u||2 * ||v||2
In think the purpose is to teach you the math. It doesn't have any relation to the real world.
 

1. What is a dot product?

A dot product, also known as a scalar product, is a mathematical operation that takes two vectors and returns a single number. It is calculated by multiplying the corresponding components of the vectors and summing the results.

2. How is a dot product used in physics?

In physics, the dot product is often used to calculate the work done by a force on an object. It is also used in calculating the angle between two vectors and determining if they are perpendicular.

3. What is a cross product?

A cross product, also known as a vector product, is a mathematical operation that takes two vectors and returns a third vector that is perpendicular to both of the original vectors. It is calculated by using the properties of determinants.

4. How is a cross product used in geometry?

In geometry, the cross product is often used to find the area of a parallelogram formed by two vectors, as well as the volume of a parallelepiped formed by three vectors. It is also used in calculating the angle between two planes.

5. What is the relationship between the dot product and the cross product?

The dot product and cross product are both mathematical operations on vectors, but they have different properties and results. The dot product returns a scalar value, while the cross product returns a vector. Additionally, the dot product is commutative, meaning the order of the vectors does not matter, while the cross product is anti-commutative, meaning the order does matter. Finally, the dot product is used for calculating the angle between two vectors, while the cross product is used for finding a vector perpendicular to two vectors.

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