The Direction Cosine Matrix: Part 1

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The Direction Cosine Matrix: Part 1

By Thomas Keyes
Dec. 24, 2005

I discussed in an article called “The Transformation of Vectors”, posted on June 27, 2005, a mathematical concept called the ‘direction cosine matrix’, which often is called alpha, but which I’ll just call A#. This also qualifies as a tensor in that it has a spatial meaning. A# is a ninefold array of cosines, arranged in three rows and three columns in the following way:

[a11, a12, a13]

[a21, a22, a23]

[a31, a32, a33]

In older books, the commas are omitted. Nowadays, with the influence of computers, they are often inserted to function as markers. Usually, the square brackets would embrace all three rows together. The designations a11, a12, a13, etc., are completely analogous to x, y and z in algebra, standing for numbers that will replace them, the only difference being that the first digit stands for the row and the second for the column.

Suppose we have nine angles, b11 through b33, which may vary from 0° to 180º: b11 = 66.5424°; b12 = 52.2744°; b13 = 133.1154°; b21 = 77.7862°; b22 = 141.8318; b23 = 125.4949°; b31 = 26.7957°; b32 = 94.9657°; b33 = 63.7416°. Then letting a11 = cos b11, a12 = cos b12, etc., and we set up the direction cosine matrix (DCM), as follows.

[.39807, .61188, -.68347]

[.21156, -.78620, -.58063]

[.89262, -.08656, .44242]

The elements of the matrix, that is, the numerical values of the cosines, are not independent. The sum of the squares of the three elements in any single row or column must equal 1. The sum of the three products of corresponding elements in any two different rows or columns must equal 0. In other words, we must have the following equalities:

a11^2 + a12^2 + a13^2 = 1

a21^2 + a22^2 + a23^2 = 1

a31^2 + a32^2 + a33^2 = 1

a11^2 + a21^2 + a31^2 = 1

a12^2 + a22^2 + a32^2 = 1

a13^2 + a23^2 + a33^2 = 1

a11 x a12 + a21 x a22 + a31 x a32 = 0

a11 x a13 + a21 x a23 + a31 x a33 = 0

a12 x a13 + a22 x a23 + a32 x a33 = 0

a11 x a21 + a12 x a22 + a13 x a23 = 0

a11 x a31 + a12 x a32 + a13 x a33 = 0

a21 x a31 + a22 x a32 + a23 x a33 = 0

Translating the above necessary and sufficient equations into numbers, we get the following answers, as we should:

.39807^2 + .61188^2 + -.68347^2 = 1

.21156^2 + -.78620^2 + -.58063^2 = 1

.89262^2 + -.08656^2 + .44242^2 = 1

.39807^2 + .21156^2 + .89262^2 = 1

.61188^2 + -.78620^2 + -.08656^2 = 1

-.68347^2 + .58063^2 + .44242^2 = 1

.39807 x .61188 + .21156 x -.78620 + .89262 x -.08656 = 0

.39807 x -.68347 + .21156 x -.58062 + .89262 x .44242 = 0

.61188 x -.68347 + -.78620 x -.58063 + -.08656 x .44242 = 0

.39807 x .21156 + .61188 x -.78620 + -.68347 x -.58063 = 0

.39807 x .89262 + .61188 x -.08656 + -.68347 x .44242 = 0

.21156 x .89262 + -.78620 x -.08656 + -.58063 x .44242 = 0

In three-dimensional space, vectors have three components. Thus we may suppose that a plane going ENE and rising at an angle of 10° is actually traveling 636.89 mph E, 263.81 mph N, and 121.55 mph up, for a speed of 700 mph, all told. Then we write V* = [636.89, 263.81, 121.55], or algebraically V* = [v1, v2, v3]. V* is called the resultant and v1, v2 and v3 are called the components. Note that Sqr [(v1)^2 + (v2)^2 + (v3)^2], which is denoted /V*/, is called the magnitude of V*, and in our example was 700. V* is a vector and has direction. /V*/ is a scalar and does not. In this case, the vector is called velocity and the scalar is called speed. In most cases, however, there are not two separate words. Thus we must say ‘vector magnetic potential’ and ‘scalar magnetic potential’.

We can multiply a vector by the DCM: V1 A# = V2. This produces another vector, identical in magnitude to the original vector, but pointed in a different direction. In Part 2, I’ll explain the method of multiplication and the application of the DCM to some other topics I’ve been discussing in recent articles.

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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

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