Direction Cosines/Examples/Example 1

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Example of Direction Cosines

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 $\dfrac {10 \sqrt 3} 3$.


Proof

From Magnitude of Vector Quantity in terms of Components:

$\size {\mathbf A} = \sqrt {x^2 + y^2 + z^2}$

where $x$, $y$ amd $z$ are the magnitudes of the components of $\mathbf A$ in the $\mathbf i$, $\mathbf j$ and $\mathbf k$ directions respectively.

From Components of Vector in terms of Direction Cosines:

\(\ds x\) \(=\) \(\ds \size {\mathbf A} \cos \alpha\)
\(\ds y\) \(=\) \(\ds \size {\mathbf A} \cos \beta\)
\(\ds z\) \(=\) \(\ds \size {\mathbf A} \cos \gamma\)

We are given that:

$\alpha = \beta = \gamma$

and that:

$\size {\mathbf A} = 10$

Hence:

\(\ds \sqrt {\paren {\size {\mathbf A} \cos \alpha}^2 + \paren {\size {\mathbf A} \cos \beta}^2 + \paren {\size {\mathbf A} \cos \gamma}^2}\) \(=\) \(\ds \size {\mathbf A}\)
\(\ds \leadsto \ \ \) \(\ds \sqrt {3 {A_x}^2}\) \(=\) \(\ds 10\) as $A_x = A_y = A_z = \size {\mathbf A} \cos \alpha$ etc.
\(\ds \leadsto \ \ \) \(\ds A_x = A_y = A_z\) \(=\) \(\ds \dfrac {10 \sqrt 3} 3\)

$\blacksquare$


Sources

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