A) \[60{}^\circ ,\text{ }60{}^\circ ,\text{ }60{}^\circ \]
B) \[45{}^\circ ,\text{ }45{}^\circ ,\text{ }45{}^\circ \]
C) \[60{}^\circ ,\text{ }60{}^\circ ,\text{ }45{}^\circ \]
D) \[45{}^\circ ,\text{ }45{}^\circ ,\text{ }60{}^\circ \]
Correct Answer: C
Solution :
\[\overrightarrow{\operatorname{R}} = \widehat{i} +\widehat{j}+ \sqrt{2}\widehat{k}\] Comparing the given vector with \[\operatorname{R}={{R}_{x}}\widehat{i}+{{R}_{y}}\widehat{j}+{{R}_{z}}\widehat{k}\] \[{{R}_{x}}=1,\,\,\,{{7}_{y}}=1,\,\,\,{{R}_{z}}=\sqrt{2}\] \[and\,\,\,\left| \overrightarrow{R} \right|\,\,=\,\,\sqrt{R_{x}^{2}+R_{y}^{2}+R_{z}^{2}}\,\,=\,\,2\] \[\cos \,\alpha \,\,=\,\,\frac{{{R}_{x}}}{R}\,=\,\frac{1}{2}\,\,\,\Rightarrow \,\,\,\alpha \,\,=\,\,60{}^\circ \] \[\cos \,\beta \,\,=\,\,\frac{{{R}_{y}}}{R}\,=\,\frac{1}{2}\,\,\,\Rightarrow \,\,\,\beta \,\,=\,\,60{}^\circ \] \[\cos \,\gamma \,\,=\,\,\frac{{{R}_{z}}}{R}\,=\,\frac{1}{\sqrt{2}}\,\,\,\Rightarrow \,\,\,\gamma \,\,=\,\,60{}^\circ \]
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