Determinants and parallelepiped

[kristo]

Homework Statement

For the determinant <br /> \left| \begin{array}{ccc} a_1 &amp; a_2 &amp; a_3 \\ b_1 &amp; b_2 &amp; b_3 \\ c_1 &amp; c_2 &amp; c_3 \end{array} \right| <br /> , b and c being the base of a parallelepiped
why is the equation \vec b \cdot (a_1^{&#039;}e_1 + a_2^{&#039;} e_2 + a_3^{&#039;} e_3) = 0 (same goes for vector c) true? Where a' is a minor of the determinant and e a unit vector.

The Attempt at a Solution

Well, it makes sense algebraically, but as to the geometrical interpretation, I don't really understand it. The vector a'e is is supposedly perpendicular to vector b and also c, but how come?

 

[kristo]

Anyone?

 

[lanedance]

consider the determinant method for calculating a cross product...

if you still don't get it try writing out what the minors are...