Changes

As we are about to see, these conveniences can make the calculation of the cross product relatively simple.  Hopefully, the student will remember the definition of the '''determinant''' of a matrix, because this will make calculating cross products much, much easier.  For those who are unfamiliar with the idea, we will review the concept.

As we are about to see, these conveniences can make the calculation of the cross product relatively simple.  Hopefully, the student will remember the definition of the '''determinant''' of a matrix, because this will make calculating cross products much, much easier.  For those who are unfamiliar with the idea, we will review the concept.

For a matrix <math>\begin{pmatrix}a&b\\c&d\end{pmatrix} \ </math>, we define the determinant <math>\begin{vmatrix}a&b\\c&d\end{vmatrix}</math> to be <math>ad - bc \ </math>.  To compute the determinant of a larger square matrix, then, we choose any row of the matrix and take the left entry, and multiply that number by the determinant of the matrix formed by eliminating the first row and column of the original matrix.  From this, we subtract the next entry in our row times the matrix formed by eliminating THAT row and column of the orginal matrix.  Then we add the next term, and so on and so forth until we have a complete expansion.   

For a matrix <math>\begin{pmatrix}a&b\\c&d\end{pmatrix} \ </math>, we define the determinant <math>\begin{vmatrix}a&b\\c&d\end{vmatrix}</math> to be <math>ad - bc \ </math>.  To compute the determinant of a larger square matrix, then, we choose any row of the matrix and take the left entry, and multiply that number by the determinant of the matrix formed by eliminating the first row and column of the original matrix.  From this, we subtract the next entry in our row times the matrix formed by eliminating THAT row and column of the original matrix.  Then we add the next term, and so on and so forth until we have a complete expansion.   

Below is an example of this technique, expanding along the first row.  

Below is an example of this technique, expanding along the first row.  

::<math>\vec{B} = \hat{\theta} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}</math>

::<math>\vec{B} = \hat{\theta} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}</math>

(The length of <math>\vec{B}</math> is r.)

(The length of <math>\vec{B}</math> is r.)

<math>\vec{B}</math> is a vector at the surface of a sphere of radius r that points "south", tangential to the surface.  <math>\vec{A}</math> points outward from the surface, that is "up".  <math>\vec{A}</math> and <math>\vec{B}</math> are orthogonal.  (Prove it; caclulate <math>\vec{A} \cdot \vec{B}</math>.)

<math>\vec{B}</math> is a vector at the surface of a sphere of radius r that points "south", tangential to the surface.  <math>\vec{A}</math> points outward from the surface, that is "up".  <math>\vec{A}</math> and <math>\vec{B}</math> are orthogonal.  (Prove it; calculate <math>\vec{A} \cdot \vec{B}</math>.)

By the formula for the cross product, we have:

By the formula for the cross product, we have: