Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Trending News
Spherical law of cosines proof?
I don't understand the following part of the proof in this article: http://en.wikipedia.org/wiki/Spherical_law_of_cosi...
"..., whose direction is given by the component of v perpendicular to u. This means:
t_a = ( v - u (u ・ v) ) / | v - u (u ・ v) | = ( v - u cos(a) ) / sin(a)
where for the denominator we have used the Pythagorean identity sin^2(a) = 1 − cos^2(a). "
Could someone provide explanations and/or intermediate steps?
2 Answers
- IndicaLv 71 decade agoFavorite Answer
Draw the two unit vectors u & v, say OU & OV. Drop the perp from V to OU to meet it at N. Then ON and NV represent the vector components of v in the direction of u and perp to u. By definition of dot product, the length of ON=u•v so ON, as a vector, is
(u•v)u. From vector addition OV = ON + NV so NV = v−(u•v)u.
However, NV is not a unit vector so for t_a we normalise.
t_a = (v−(u•v)u) / |v−(u•v)u|
Further, ∠VON=α so length of ON=cos(α) and length of NV=sin(α)
∴ t_a = (v −cos(α)u) / sin(α)
