Vectors and Matrices

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+ Next: Using vectors +
+Up: EXPLANATION AND EXAMPLES +
+ Previous: Formatting angles +

+Vectors and Matrices +

+As an alternative to employing a spherical polar coordinate system, +the direction of an object can be defined in terms of the sum of any +three vectors as long as they are different and +not coplanar. In practice, three vectors at right angles are +usually chosen, forming a system +of Cartesian coordinates. The x- and y-axes +lie in the fundamental plane (e.g. the equator in the +case of $[\,\alpha,\delta\,]$ ), with the x-axis pointing to zero longitude. +The z-axis is normal to the fundamental plane and points +towards positive latitudes. The y-axis can lie in either +of the two possible directions, depending on whether the +coordinate system is right-handed or left-handed. +The three axes are sometimes called +a triad. For most applications involving arbitrarily +distant objects such as stars, the vector which defines +the direction concerned is constrained to have unit length. +The x-, y- and z-components +can be regarded as the scalar (dot) product of this vector +onto the three axes of the triad in turn. Because the vector +is a unit vector, +each of the three dot-products is simply the cosine of the angle +between the unit vector and the axis concerned, and the +x-, y- and z-components are sometimes +called direction cosines. +

+For some applications involving objects +with the Solar System, unit vectors are inappropriate, and +it is necessary to use vectors scaled in length-units such as +AU, km etc. +In these cases the origin of the coordinate system may not be +the observer, but instead might be the Sun, the Solar-System +barycentre, the centre of the Earth etc. But whatever the application, +the final direction in which the observer sees the object can be +expressed as direction cosines. +

+But where has this got us? Instead of two numbers - a longitude and +a latitude - we now have three numbers to look after +- the x-, y- and +z-components - whose quadratic sum we have somehow to contrive to +be unity. And, in addition to this apparent redundancy, +most people find it harder to visualize +problems in terms of $[\,x,y,z\,]$ than in $[\,\theta,\phi~]$ .Despite these objections, the vector approach turns out to have +significant advantages over the spherical trigonometry approach: +

Vector formulae tend to be much more succinct; one vector + operation is the equivalent of strings of sines and cosines. +
The formulae are as a rule rigorous, even at the poles. +
Accuracy is maintained all over the celestial sphere. + When one Cartesian component is nearly unity and + therefore insensitive to direction, the others become small + and therefore more precise. +
Formulations usually deliver the quadrant of the result + without the need for any inspection (except within the + library function ATAN2). +

+A number of important transformations in positional +astronomy turn out to be nothing more than changes of coordinate +system, something which is especially convenient if +the vector approach is used. A direction with respect +to one triad can be expressed relative to another triad simply +by multiplying the $[\,x,y,z\,]$ column vector by the appropriate + $3\times3$ orthogonal matrix +(a tensor of Rank 2, or dyadic). The three rows of this +rotation matrix +are the vectors in the old coordinate system of the three +new axes, and the transformation amounts to obtaining the +dot-product of the direction-vector with each of the three +new axes. Precession, nutation, $[\,h,\delta\,]$ to $[\,Az,El~]$ , $[\,\alpha,\delta\,]$ to $[\,l^{I\!I},b^{I\!I}\,]$ and so on are typical examples of the +technique. A useful property of the rotation matrices +is that they can be inverted simply by taking the transpose. +

+The elements of these vectors and matrices are assorted combinations of +the sines and cosines of the various angles involved (hour angle, +declination and so on, depending on which transformation is +being applied). If you write out the matrix multiplications +in full you get expressions which are essentially the same as the +equivalent spherical trigonometry formulae. Indeed, many of the +standard formulae of spherical trigonometry are most easily +derived by expressing the problem initially in +terms of vectors. +

+ + +

+Using vectors +

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+
+ Next: Using vectors +
+Up: EXPLANATION AND EXAMPLES +
+ Previous: Formatting angles +

+SLALIB --- Positional Astronomy Library
Starlink User Note 67
P. T. Wallace
12 October 1999
E-mail:ptw@star.rl.ac.uk +

+ + -- cgit