Definition:Direction Cosines

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Definition

Let $\mathbf a$ be a vector quantity embedded in a Cartesian $3$-space.

Let the angles which $\mathbf a$ makes with the $x$-axis, $y$-axis and $z$-axis be $\alpha$, $\beta$ and $\gamma$ respectively.


Then the direction cosines of $\mathbf a$ are $\cos \alpha$, $\cos \beta$ and $\cos \gamma$, defined individually such that:

$\cos \alpha$ is the direction cosine of $\mathbf a$ with respect to the $x$-axis
$\cos \beta$ is the direction cosine of $\mathbf a$ with respect to the $y$-axis
$\cos \gamma$ is the direction cosine of $\mathbf a$ with respect to the $z$-axis.


Vector-Components-in-3-Space.png


Also presented as

Some sources do not dwell on the actual angles themselves, but instead denote the direction cosines directly as $\alpha$, $\beta$ and $\gamma$.

While this technique results in more streamlined notation, it can result in confusion.


Examples

Example

Let $\mathbf A$ be a vector quantity of magnitude $10$ embedded in Cartesian $3$-space.

Let $\mathbf A$ make equal angles with the coordinate axes $x$, $y$ and $z$.


Then the magnitudes of the components of $\mathbf A$ are all equal to $\dfrac {10 \sqrt 3} 3$.


Also see

  • Results about direction cosines can be found here.


Sources

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