Hi,
This is my first post on the forum, but I have been reading this forum for at least 2 years now. It has always been very useful to read some of the posts you guys make.
However, now, it seems like I can't solve my problem simply by reading the forum...I would need a little extra help.
Here is how it goes:
(Note: Let bold letters be vectors)
I know that u=(u1,u2,u3), v=(v1,v2,v3) and w=(w1,w2,w3)
Let A=
I have to prove that:
(Note: Let u,v and w, be vectors)
det(A)
I tried to replace the norm of u times norm of v times norm of w by its equivalent using their values (u1,u2,u3), (v1,v2,v3), (w1,w2,w3).
Hence, ...
But then, all I got was a huge equation and the strange impression that I am not even close from doing the right thing...
Anyone could help me head in the right direction ?
Thanks a lot !
When you cross two vectors you get a vector that is normal(perpendicular) to the plane spanned by the two vectors.
The n is just giving you the direction that the resulting vector is going.
The is the length(magnitude) of the resulting vector.
This is needed becuase the dot product is a binary operation on two vectors, not a vector and magnitude. If we didn't have the unit normal the dot product would not make sense.
I hope this helps.
I have editied the above post for clarity as well.
Is incorrect. a is just the vector a.
Now:
where is the angle beteween d and e.
Therefore:
Now n is a unit vector so:
Be careful, don't confuse and together.
The one inside sin is the angle between b and c, and the one inside cos is the angle between a and the resulting vector of (bxc).
Ok,
So basically,
is equal to Det(A)...but how does this prove that the det(A) is smaller than
Edit: Oh, is it simply that and will always give values bellow 1 (or equal to one), and therefore the value of times 2 values that are equal or smaller to 1 will always give a smaller or equal value than
is a vector while is a length of the vector.
The length of the vector that is the result of a cross product is
,
but the vector it self is
Read again what TheEmptySet posted above:
When you cross two vectors you get a vector that is normal(perpendicular) to the plane spanned by the two vectors.
The n is just giving you the direction that the resulting vector is going.
The is the length(magnitude) of the resulting vector.
This is needed becuase the dot product is a binary operation on two vectors, not a vector and magnitude. If we didn't have the unit normal the dot product would not make sense.
I hope this helps.
I have editied the above post for clarity as well.