Understanding the Relationship between Dot and Cross Products

  • #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.
 
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