Question

Why does the determinant of vectors A B C =
|a1 a2 a3| |b2 b3| |b1 b3| |b1 b2|
|b1 b2 b3| = a1|c2 c3| - a2|c1 c3|+a3|c1 c2| ?
|c1 c2 c3|

Answer 1

this determinant is the expression for solving the scalar triple product of the three vectors given as:
\[\vec{a}\cdot(\vec{b}\times\vec{c})=\left|\begin{matrix}
a_1&a_2&a_3\\
b_1&b_2&b_3\\
c_1&c_2&c_3\\
\end{matrix}\right|=a_1\left|\begin{matrix}b_2&b_3\\c_2&c_3\end{matrix}\right|
-a_2\left|\begin{matrix}b_1&b_3\\c_1&c_3\end{matrix}\right|
+a_3\left|\begin{matrix}b_1&b_2\\c_1&c_2\end{matrix}\right|
\]
the smaller determinants, you can solve, right?

Answer 2

yes, but why is this so? why is a•(b x c) = det(a,b,c) ? I guess to be more specific what is the geometric meaning of these two ways of expressing it.

Answer 3

\[\vec{b}\times\vec{c}=\vec{d}\]
it is a vector.. perpendicular to the plane of b and c
now, a dot product will give you a scalar, which is the projection of a on d