Vector Algebra

A vector $\overrightarrow{r}$ has length $35 \sqrt{2}$ and direction ratios $\left(3, 4, 5\right)$. Find the direction cosines and the components of $\overrightarrow{r}$, assuming that $\overrightarrow{r}$ makes an acute angle with the $X$ axis.


Given: $\;$ $\left|\overrightarrow{r}\right| = 35 \sqrt{2}$

Let $\ell$, $m$ and $n$ be the direction cosines of $\overrightarrow{r}$.

$\because$ $\;$ $\overrightarrow{r}$ makes an acute angle with the $X$ axis, $\ell > 0$.

Direction ratios are $\;$ $3$, $4$, $5$.

We have $\;$ $\sqrt{\left(3\right)^2 + \left(4\right)^2 + \left(5\right)^2} = \sqrt{50} = 5 \sqrt{2}$

$\therefore$ $\;$ direction cosines of $\overrightarrow{r}$ are $\;$ $\ell = \dfrac{3}{5 \sqrt{2}}$, $\;$ $m = \dfrac{4}{5 \sqrt{2}}$, $\;$ $n = \dfrac{5}{5 \sqrt{2}}$

Now,

$\begin{aligned} \overrightarrow{r} & = \left|\overrightarrow{r}\right| \left(\ell \hat{i} + m \hat{j} + n \hat{k}\right) \\\\ & = 35 \sqrt{2} \left(\dfrac{3}{5 \sqrt{2}} \hat{i} + \dfrac{4}{5 \sqrt{2}} \hat{j} + \dfrac{5}{5 \sqrt{2}} \hat{j}\right) \\\\ & = 21 \hat{i} + 28 \hat{j} + 35 \hat{k} \end{aligned}$

$\therefore$ $\;$ Components of $\overrightarrow{r}$ along the $X$, $Y$ and $Z$ axes are: $\;$ $21 \hat{i}$, $28 \hat{j}$ and $35 \hat{k}$ respectively.