Find the angles

[hbxdz]

Hi , i got a homework for my summer classes (algebra) and i don't get it.
I don't understand how to start when they only give me the air.

A tetrahedron is defined by the vectors u , v & w where w=(2,2,1).
knowing that (u ∧ v)*w=18 and the tetrahedron air of u and v is 6 ,find the angle between (u ∧ v) and w.


(Read the picture as the vector u is at the bottom)


thank you !!!

 

[Dr.Peterson]

Please explain the notation you are using. What do ∧ and * mean in your context? And what is "tetrahedron air"??

 

[hbxdz]

∧ means the vector product and * means the multiplication of (u ∧ v) and w
the tetrahedron air is the air of the triangle u and v

 

[Dr.Peterson]

Do you mean AREA, not "air"? (Air is the gas we breathe.) And why would you call it "tetrahedron area" if it pertains only to one face of it?

In any case, you are apparently talking about the scalar triple product, which I would write as [MATH]u\times v)\cdot w[/MATH], which gives the volume of a parallelepiped, 6 times the volume of the tetrahedron. That's what I suspected you might mean.

The vector product is a vector normal to the plane with magnitude equal to the area of the parallelogram base (therefore twice the triangle area); and the scalar product is related to the angle between that and vector w.

Does that remind you of enough facts to solve the problem?

 

[hbxdz]

Yes exactly it was the ''area'' !
So if the vector product is equal to the area of the parallelogram it means that it's equal to 12 ?
And the 18 i don understand how to use it in an equation ?

 

[Dr.Peterson]

Do you know the formula for the scalar (dot) product in terms of magnitudes and the angle?