# Thornton & Marion, Classical Dynamics of Particles and Systems, 5th Edition

## Chapter 1. Matrices, Vectors, and Vector Calculus

### Problem 02. Trigonometric properties of direction cosines

#### Solution

For Equation (1.10): $latex \\cos^2\\alpha + \\cos^2\\beta + \\cos^2\\gamma = 1$ Proof)

Using Figure 1-4 (a) in the book, let the length of a point P on the line $latex (\\alpha, \\beta, \\gamma)$ from the origin be $latex l$, then

$latex l^2\\cos^2\\alpha + l^2\\cos^2\\beta + l^2\\cos^2\\gamma = l^2$

Therefore, it becomes

$latex \\cos^2\\alpha + \\cos^2\\beta + \\cos^2\\gamma = 1$

For Equation (1.11): $latex \\cos\\alpha\\cos\\alpha\’ + \\cos\\beta\\cos\\beta\’ + \\cos\\gamma\\cos\\gamma\’ = \\cos\\theta$

In Figure 1-4 (b) in the book, let the length of a point P on the line $latex (\\alpha, \\beta, \\gamma)$ from the origin be $latex l$ and the length of a point P\’ on the line $latex (\\alpha\’, \\beta\’, \\gamma\’)$ be $latex l\’$.

$latex l = \\sqrt{l^2\\cos^2\\alpha + l^2\\cos^2\\beta + l^2\\cos^2\\gamma}$
$latex l\’ = \\sqrt{l\’^2\\cos^2\\alpha\’ + l\’^2\\cos^2\\beta\’ + l\’^2\\cos^2\\gamma\’}$

Using the law of cosines,

$latex \\overline{PP\’}^2 = l^2 + l\’^2 – 2 l l\’ \\cos\\theta$

Since $latex \\begin{array}{rcl} \\overline{PP\’}^2 & = & (l\\cos\\alpha – l\’\\cos\\alpha\’)^2 + (l\\cos\\beta – l\’\\cos\\beta\’)^2 + (l\\cos\\gamma – l\’\\cos\\gamma\’)^2 \\\\ & = & l^2(\\cos^2\\alpha + \\cos^2\\beta + \\cos^2\\gamma) + l\’^2(\\cos^2\\alpha\’ + \\cos^2\\beta\’ + \\cos^2\\gamma\’) – 2 l l\’ (\\cos\\alpha\\cos\\alpha\’ + \\cos\\beta\\cos\\beta\’ + \\cos\\gamma\\cos\\gamma\’) \\\\ & = & l^2 + l\’^2 – 2 l l\’ (\\cos\\alpha\\cos\\alpha\’ + \\cos\\beta\\cos\\beta\’ + \\cos\\gamma\\cos\\gamma\’) \\end{array}$ the equation becomes

$latex l^2 + l\’^2 – 2 l l\’ (\\cos\\alpha\\cos\\alpha\’ + \\cos\\beta\\cos\\beta\’ + \\cos\\gamma\\cos\\gamma\’) = l^2 + l\’^2 – 2 l l\’ \\cos\\theta$

Therefore,

$latex \\cos\\alpha\\cos\\alpha\’ + \\cos\\beta\\cos\\beta\’ + \\cos\\gamma\\cos\\gamma\’ = \\cos\\theta$

#### Reference

https://en.wikipedia.org/wiki/Law_of_cosines
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.study-astrophysics.com/2018/01/19/marion-01-02/", "fetch_time": 1718631075000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-26/segments/1718198861719.30/warc/CC-MAIN-20240617122241-20240617152241-00606.warc.gz", "warc_record_offset": 876290208, "warc_record_length": 16273, "token_count": 829, "char_count": 2107, "score": 4.5625, "int_score": 5, "crawl": "CC-MAIN-2024-26", "snapshot_type": "latest", "language": "en", "language_score": 0.315481960773468, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00040-of-00064.parquet"}