Understanding the Relationship between Dot and Cross Products

[zr95]

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?

 

[Hornbein]

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.

 

[zr95]

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.

 

[Hornbein]

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.

 

[zr95]

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

 

[haruspex]

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.

 

[Hornbein]

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.