The direction cosines of $\vec{r}$ are $\frac{2}{7}, -\frac{3}{7}, \frac{6}{7}$, and its $\hat{k}$ component is 12. What is $\lvert \vec{r} \rvert$?

Explanation

1. 01

   Each component is the magnitude times its direction cosine: $r_3 = \frac{6}{7}\lvert \vec{r} \rvert$.

2. 02

   $12 = \frac{6}{7}\lvert \vec{r} \rvert$, so $\lvert \vec{r} \rvert = 14$.