# three vectors are linearly independent

• Oct 17th 2009, 11:43 AM
Frostking
three vectors are linearly independent
I am suppose to use the standard volume interpretation of the scalar triple product a dot b crossed with c

where a, b and c are my vectors to support that these vecors are independent. I realize that their determinant matrix is not zero and that is what defines independence, but I do not know what the question is asking me to do!!! Any guidance would be much appreciated. Frostking
• Oct 17th 2009, 07:20 PM
redsoxfan325
Quote:

Originally Posted by Frostking
I am suppose to use the standard volume interpretation of the scalar triple product a dot b crossed with c

where a, b and c are my vectors to support that these vecors are independent. I realize that their determinant matrix is not zero and that is what defines independence, but I do not know what the question is asking me to do!!! Any guidance would be much appreciated. Frostking

If $\displaystyle a\cdot(b\times c)=0$, then the volume of the parallelepiped formed by the three vectors is $\displaystyle 0$. But that means that $\displaystyle a$, $\displaystyle b$, and $\displaystyle c$ all lie in the same plane and are therefore not linearly independent. (Since a plane is spanned by exactly two vectors, a third vector in the plane must be a linear combination of the other two.)