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A vector $\vec{v}$ of magnitude $v$ makes an angle $\alpha$ with the positive x-axis, angle $\beta$ with positive y axis, and angle $\gamma$ with the positive z-axis. Show that

$$\vec{v} = vcos(\alpha)\vec{i} + vcos(\beta)\vec{j} + vcos(\gamma)\vec{k} $$

$cos(\alpha), cos(\beta)$ and $cos(\gamma)$ are called direction cosines

Show that

$$cos^2(\alpha)+cos^2(\beta)+cos^2(\gamma)=1$$

I think this might include the geometric expression of the dot product

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  • $\begingroup$ Yep, it does! What're two ways of writing $\vec{v} \cdot \vec{v}$? (Hint: distribute) $\endgroup$
    – Henry Swanson
    Nov 6, 2014 at 5:59

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By the hypothesis the unit vector $\frac{\vec v}{v}=: x\vec i+y\vec j+z\vec k$ make an angle $\alpha$ with the positive $x$-axis so $$x=\frac{\vec v}{v}\cdot \vec i=\cos\alpha$$ and by the same method we find $y=\cos \beta$ and $z=\cos \gamma$ so we get $\vec v$. Finally notice that $$v=||\vec v||=v\sqrt{\cos^2\alpha+\cos^2\beta+\cos^2\gamma}$$

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