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
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $1$ Vector Analysis $1.1$ Definitions, Elementary Approach: Exercise $1.1.2$