» Required Reading » Table of Contents » Chapter 4 4.6 The vector product
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The vector product of two 3-vectors, v and w, written as vw is the determinant of the 3 by 3 matrix whose first  two columns are the components of v and w and whose third column consists of the basis vectors i, j and k.

It is a vector.

Its component in the k direction is the area of the parallelogram in the xy plane determined by the x and y coordinates of v and w, namely v_xw_y – v_yw_x’.

Its other components can be obtained by cyclically shifting among the variables x,y and z changing x to y, y to z and z to x.

The vector product is, from its definition, linear in both its variables, and changes sign if their order is reversed. Its magnitude is the area of the parallelogram defined by its columns or rows (that is, and it is a vector, not a number that is perpendicular to both its arguments.

We therefore have (vw) (vw) = (vv)(ww) – (vw)(wv) since both represent the square of the area of the parallelogram with sides v and w.

Note: human beings make numerical errors in evaluating 3 by 3 determinants or vector products roughly one out of every four times they do them, even more so when the vectors components or matrix entries have lots of negative values. It is wise to build a determinant and vector product tool on a spreadsheet and use it to check not replace your own computations. You will then get the right answer every time.

Exercises:

4.1,2 and 3 Construct spreadsheets that compute determinants in 2 3 and 4 dimensions.

4.4 Construct a spreadsheet that constructs the vector product of two input vectors in 3d.

4.5 Suppose you take the 2 by 2 matrix A with rows (2, 3) and (1, 4) and multiply it by the column  2 vector (cos, sin)  Use the spreadsheet you created in exercise 3.7i here to find values of for which the ratio of second coordinate to first in the product is tan, which is the same ratio as for the input vector (you can plot for different values and try to home in on it).

4.6 Do the same with the applet that does this kind of thing.

4.7 Write the characteristic equation for this matrix, solve it for c, and find the corresponding eigenvectors each up to a constant. What is their ratio of second coordinate to first in each?

4.8  Suppose you know that (vw) = A and vw = u for known vectors v and u. Can you determine w? If so what is it?

4.9 Suppose you have three 4-vectors w v and u. How might you find the volume of the parallelepiped they determine? (one conceivable way is to define a 4 dimensional analogue of the vector product (as it is defined above) and compute its magnitude as the square root of its dot product with itself. Will this work?)

To see the vector product in action, examine the applet in section 4.1.