# Thread:

1. ## Angle Between Two Vectors

Greetings,

I would like to ask whether the cosine of the angle between two vectors can be derived from the knowledge of directional cosines. If that is the case, how can I prove the statement? I am attaching a scanned page from the book Vector and Tensor Analysis by Hay, which illustrates the concept beginning with the phrase, "By a formula from of analytic geometry..." and writes the formula, around which I have put a red box, below this statement. I have put a red rectangular box to emphasize my point.

Thanks
Attached Thumbnails

2. ## Re: Angle Between Two Vectors

The theorem in the red box is pretty much a restatement of (7.1). Not particularly helpful.

To find $\cos \theta$, draw vectors $\vec{a}$ and $\vec{b}$, along with their difference, $\vec{a} - \vec{b}$, forming a triangle. Note that the dot product obeys the distributive property, that is

$(\vec{a} - \vec{b})^2 = (\vec{a})^2 + (\vec{b})^2 - 2(\vec{a} \cdot \vec{b}) = ||\vec{a}||^2 + ||\vec{b}||^2 - 2(\vec{a} \cdot \vec{b})$. However by the law of cosines,

$(\vec{a} - \vec{b})^2 = ||\vec{a}||^2 + ||\vec{b}||^2 - 2||a|| \hspace{1 mm}||b|| \cos \theta$

Set these two expressions equal, cancel stuff, and you will obtain $\vec{a} \cdot \vec{b} = ||\vec{a}||\hspace{1 mm}||\vec{b}|| \cos \theta$, or $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{||\vec{a}||\hspace{1 mm}||\vec{b}||}$

3. ## Re: Angle Between Two Vectors

Thanks, you are right, but I think you emphasize that no such formula exists in analytical geometry if I am not mistaken. However, I still wonder why the author wrote, "By a formula of analytical geometry..." do you have an explanation?
{"extraction_info": {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9356836676597595, "perplexity_flag": "head"}