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lecture_19_slides
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| lecture_19_slides [2017/04/10 16:19] – [Corollary: the length of $\vec v\times\vec w$] rupert | lecture_19_slides [2017/04/11 09:56] (current) – [Corollary: the length of $\vec v\times\vec w$] rupert | ||
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| For any vectors $\vv$ and $\ww$ in $\mathbb{R}^3$, | For any vectors $\vv$ and $\ww$ in $\mathbb{R}^3$, | ||
| \[ \|\vv\times\ww\|=\|\vv\|\, | \[ \|\vv\times\ww\|=\|\vv\|\, | ||
| - | where $\theta$ is the angle between $\vv$ and $\ww$ (with $0\le\theta<\pi$). | + | where $\theta$ is the angle between $\vv$ and $\ww$ (with $0\le\theta\le\pi$). |
| === Proof === | === Proof === | ||
| Line 116: | Line 116: | ||
| * $=\|\vv\|^2\, | * $=\|\vv\|^2\, | ||
| * $=\|\vv\|^2\, | * $=\|\vv\|^2\, | ||
| - | * $\sin\theta\ge0$ for $0\le\theta<\pi$, so taking square roots of both sides gives $ \|\vv\times\ww\|=\|\vv\|\, | + | * $\sin\theta\ge0$ for $0\le\theta\le\pi$, so taking square roots of both sides gives $ \|\vv\times\ww\|=\|\vv\|\, |
| ===== Geometry of the cross product ===== | ===== Geometry of the cross product ===== | ||
lecture_19_slides.1491841184.txt.gz · Last modified: by rupert