V3-Determinant as Related to the Right Hand Rule

Determinant and Vector Calculus

The determinant of a `3 times 3` matrix
     `A = ((a_1, a_2, a_3), (b_1, b_2, b_3),(c_1, c_2, c_3))`

is the number `a_(1)*(a_(2) *c_(3) - b_(3)*c_(2))+ a_(2)*(b_(3)*c_(1) - b_(1)*c_(3)) +  a_(3)*(b_(1)*c_(2) - b_(2)*c_(1)) .`
It is denoted `det A`. 

In Vector Calculus, the determinant of a `3 times 3` matrix corresponds to the supposition of three vectors of `bbb"R"^3` in standard position `vec(a) = (a_(1), a_(2), a_(3)),` `vec(b) = (b_(1), b_(2), b_(3)),` and `vec(c) = (c_(1), c_(2), c_(3)),` whereby we consider the matrix
     `A = ((a_1, a_2, a_3), (b_1, b_2, b_3),(c_1, c_2, c_3))`
that is formed by using the coordinates of each respective vector to make a row of the matrix (Bray, 25-28). 

There are three options that the calculated determinant might fall under (Bray, 25-28): 

• The value of the determinant may be positive, which means that the vectors are in counterclockwise order. 
• The value of the determinant may be negative, which means that the vectors are in clockwise order. 
• The value of the determinant is zero, and the vectors are parallel to each other.

This relates to the right hand rule because the right hand rule is used as a way to check if vectors in three-dimensions are in counterclockwise order; in which case the sign of the determinant of a `3 times 3` matrix is another indicator of this directionality. Right-handed versus left-handed coordinates, which are analogous to the right hand rule, are described by Wikipedia as follows:

"For right-handed coordinates your right thumb points along the `Z` axis and the curl of your fingers represents a motion from the first or `X` axis to the second or `Y` axis. When viewed from the top or `Z` axis the system is counter-clockwise.

For left-handed coordinates your left thumb points along the `Z` axis and the curled fingers of your left hand represent a motion from the first or `X` axis to the second or `Y` axis. When viewed from the top or `Z` axis the system is clockwise."

Interestingly and sensibly, the determinant of the matrix `A` also corresponds to the area of a parallelepiped formed with one vertex at the origin, one vertex at the point defined by the sum of all three vectors, and three of the edges defined by the vectors. The volume of this defined parallelepiped is equal to the absolute value of the determinant of matrix `A`. Of course, we note that if the determinant is zero then the volume of the parallelepiped is also zero (Bray, 25-28). 

For a `2 times 2` matrix, the sign of the determinant of the matrix formed by the vectors also indicates directionality; the absolute value of the determinant, however, corresponds to the area of a parallelogram formed by the vectors. 

Sources

Bray, Clark. Multivariable Calculus. Middletown, DE: n.p., 2015. Print.
"Right-hand Rule." Wikipedia. Wikimedia Foundation, n.d. Web. 09 June 2016. 

See Also

https://en.wikipedia.org/wiki/Right-hand_rule