A vector $\vec{r}$ has magnitude 14 and direction ratios $2, 3, –6$. Find the direction cosines and components of $\vec{r}$ , given that $\vec{r}$ makes an acute angle with x-axis.
The solution given in my reference is: $(l,r,m)=\big(\frac{2}{7},\frac{3}{7},\frac{-6}{7}\big)$ and components are $4\hat{i}, 6\hat{j}, -12\hat{k}$
My Doubt $$ |\vec{r}|=14\neq \sqrt{4+9+36}=7 $$
So, are the terms direction ratios and scalar components not the same ?
My Understanding
$$ \vec{r}=\Big(x\hat{i}+y\hat{j}+z\hat{k}\Big)=|\vec{r}|\Big(l\hat{i}+m\hat{j}+n\hat{k}\Big)=|\vec{r}|\Big(\cos\alpha\hat{i}+\cos\beta\hat{j}+\cos\gamma\hat{k}\Big) $$ where $x=l|\vec{r}|, y=m|\vec{r}|, z=n|\vec{r}|$ are the direction ratios.