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# A vector $\overrightarrow{r}$ has length $35\sqrt{2}$ and direction ratios $(3 , 4 , 5 ),$ Find the direction consines and components of $\overrightarrow{r}$.

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• The d.c.s of a vector with d.r.s $(a,b,c)$ are $\large\frac{a}{\sqrt{a^2+b^2+c^2}}, \large\frac{b}{\sqrt{a^2+b^2+c^2}}, \large\frac{c}{\sqrt{a^2+b^2+c^2}}$
The d.c.s of the vector are $\large\frac{3}{\sqrt{9+16+25}},\large\frac{4}{\sqrt{9+16+25}},\large\frac{5}{\sqrt{9+16+25}}$
$= \large\frac{3}{\sqrt{50}},\large\frac{3}{\sqrt{50}},\large\frac{5}{\sqrt{50}}= \large\frac{3}{5\sqrt 2}, \large\frac{4}{5\sqrt 2}\large\frac{5}{5\sqrt 2}$
The required vector is of length $35\sqrt 2$
$\therefore \overrightarrow r = 35\sqrt 2 \bigg( \large\frac{3}{5\sqrt 2} \overrightarrow i + \large\frac{4}{5\sqrt 2} \overrightarrow j + \large\frac{5}{5\sqrt 2} \overrightarrow k \bigg)$
$= 7(3\overrightarrow i + 4 \overrightarrow j + 5\overrightarrow k)$

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