Chapter 1: Vectors, Lines and Planes
Section 1.3: Dot Product
Example 1.3.14
If A and B are unit vectors with θ the angle between them, show that sinθ/2=12A−B.
Solution
To establish the given identity, begin by squaring both sides. The left side then yields to the half-angle formula for the sine; the right side contains the square of the norm of the difference, which then becomes the dot product of the difference with itself. Each time the norm of A or B appears, remember that these are unit vectors, so the norms are 1.
sin2θ/2
=14A−B2
1−cosθ2
=14A−B·A−B
=14A·A+B·B−2 A·B
=14(A2+∥B∥2−2 A·B)
=142−2 A·B
=121−A·B
=1− A B cos(θ)2
=1−cosθ2
<< Previous Example Section 1.3 Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
