HOME | POLITICS | SPORTS | LIFE | SCI/TECH | OPEDS | HELPFUL TIPS
|
Dec. 27, 2005 In Part 1, I displayed a randomly chosen direction cosine matrix (DCM), which now I’ll call A1#, where I am using the pound sign to signify a three-by-three matrix, which in this case is also a second-rank tensor. Not all matrices are tensors. For example, simultaneous equations involving apples, oranges and lemons can be solved by using matrices, but such matrices would not be called tensors, which, in ordinary mechanics, have a spatial meaning. Now let me create a DCM for a comprehensible situation. Suppose we start by setting up a coordinate system. We draw three lines, the first line east-west, the second one north-south and the third one up-down, all intersecting in a point called the origin. The three lines are our XX, YY and ZZ axes. Suppose we have our axes dead center in some sort of chamber or compartment. Now every point in the chamber can be located by listing its coordinates, always in the same order, first from XX, next from YY, and finally from ZZ. If the first coordinate is east it is positive; if it is west it is negative. If the second is north, it is positive; if it is south it is negative. If the third is up, it is positive; if it is down it is negative. A set of three such numbers is called the position vector or radius vector of a point. There are basically two different ways to display the components of the vector. Let’s suppose our point is 6.4 meters east, 8 meters north and 3 meters down, with respect to the origin. Then in one system of notation, we have: V1* = [6.4, 8, -3]. In the other, we have V1* =6.4i + 8j - 3k. Here i,j, and k denote unit vectors directed east, north and up. Obviously, we may identify as many such vectors as we choose, calling them V2*, V3*, V4*, etc. Note that I have instanced meters merely as an example. Any of a number of simple or compound units of measure might be applicable in a given case. Now suppose we move the whole chamber or compartment, including all the points that we have labeled. We turn it counterclockwise as we look down from above so that the XX axis is now directed ENE, that is 22.5° north of east. Consequently, the YY axis will be directed due NNW, still perpendicular to XX. The ZZ axis remains unaffected. Now let me rotate the chamber a second time, turning it counterclockwise about the new YY axis 10°, tilting the new XX axis down towards the positive end and the new ZZ so that its positive end inclines a little towards WSW, while the new YY remains unaffected. By means of trigonometry and algebra, I can deduce the DCM for the new position of the chamber, which I’ll just call A2#: [.92178, .37770,-.08750] [-.38703, .90979, -.14999] [.02294,.17213,.98481] Multiplying the vector V1* by the DCM A2# gives us a new vector, which we may just call V1*’: V1* A2# = V1*’. To execute the multiplication, we first multiply each of the three components of the vector by the corresponding component in the first column of the matrix, adding the three products. Then we multiply them by the corresponding components of the second column of the matrix, adding the products. And we do the same with the third column. Explicitly we have: v1 x a11 + v2 x a21 + v3 x a31 = v1’ v1 x a12 + v2 x a22 + v3 x a32 = v2’ v1 x a13 + v2 x a23 + v3 x a33 = v3’ The new vector V1*’ will equal [v1’, v2’, v3’] This multplication can be done item by item with any calculator, but some calculators, like TI-85, can do it instantanteously. Online matrix multiplication can be done at the following URL, among others: http://wims.unice.fr/wims/wims.cgi At that website, you simply click on Matrix Multiplier. Then you enter a vector in the space called ‘A =’ thus: 6.4, 8, -3 The matrix is entered in the space called ‘B =’ thus: .92178, .37770,-.08750 -.38703, .90979, -.14999 .02294,.17213,.98481 In the box called ‘C =’, you type in: A*B Then you click on ‘evalue’, and your answer appears above: C = (2.734332, 9.17291, -4.71435) In the notation I have been using, therefore, we would say V1*’ = [2.734332,9.17291,-4.71435]. These are the coordinates in the old system of the point that was originally at [6.4,8,-3] in the old system and remains [6.4,8,-3] in the new system when the specified rotations have been made. ------------ About the author Thomas Keyes: I have written two books: A SOJOURN IN ASIA (non-fiction) and A TALE OF UNG (fiction), neither published so far. I have studied languages for years and traveled extensively on five continents. Email: udikeyes@yahoo.com Tell a friend about this site! ------------ All articles are EXCLUSIVE to Useless-Knowledge.com and are not allowed to be posted on other websites. ARTICLE THIEVES WILL BE PROSECUTED! |
||||||
|
|
|||||||
|
