Direction Cosines/Examples/Example 1

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:

We are given that:
 * $\alpha = \beta = \gamma$

and that:
 * $\size {\mathbf A} = 10$

Hence: