# Re: Vector Calc

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Posted by Soroban on September 09, 2002 at 11:54:07:

In Reply to: Vector Calc posted by Ccp511 on September 09, 2002 at 02:37:45:

: if v X w= 2i-3j+5k, and v.w=3, find tan(theta) where theta is the angle between vectors v and w. What is the general formula that relates dot product and cross product and how can I use that to find the angle between the two vectors?

What a devious problem!
I don't think there is a "general formula" that relates Dot product
to Cross Product, but we can invent one!

I had to dig through the cobwebs in my brain to come up with this.
See what you think.

We have two vectors V and W.

Recall that the area of the parallelogram determined by V and W
is given by: A = |V x W|.

Also, the area of that parallelogram is: A = |V||W|sin @
where @ is the angle between the vectors.

So, |V||W|sin @ = |VxW|. Hence, sin @ = |VxW|/|V||W|.

Now, the angle between V and W is: cos @ = V*W/|V||W|.

Divide the last two equations:
(sin @)/(cos @) = |VxW|/V*W

We got it! We find that: |VxW| = sqrt(38),
so: tan @ = sqrt(38)/3

I got 64.05 degrees.

[Before I shout "It's Miller time!", is this a
well-known derivation? Is it a centuries-old
formula? Did I just "reinvent the wheel"?]

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