Question 62: Motion in 2D & Vectors

Question

The magnitude of vectors $$\overrightarrow {OA} $$, $$\overrightarrow {OB} $$ and $$\overrightarrow {OC} $$ in the given figure are equal. The direction of $$\overrightarrow {OA} $$ + $$\overrightarrow {OB} $$ $$-$$ $$\overrightarrow {OC} $$ with x-axis will be :

Accepted Answer

Let magnitude be equal to $$\lambda$$
$$\overrightarrow {OA}  = \lambda \left[ {\cos 30^\circ \widehat i + \sin 30\widehat j} ight] = \lambda \left[ {{{\sqrt 3 } \over 2}\widehat i + {1 \over 2}\widehat j} ight]$$
$$\overrightarrow {OB}  = \lambda \left[ {\cos 60^\circ \widehat i - \sin 60\widehat j} ight] = \lambda \left[ {{1 \over 2}\widehat i - {{\sqrt 3 } \over 2}\widehat j} ight]$$
$$\overrightarrow {OC}  = \lambda \left[ {\cos 45^\circ ( - \widehat i) + \sin 45\widehat j} ight] = \lambda \left[ { - {1 \over {\sqrt 2 }}\widehat i + {1 \over {\sqrt 2 }}\widehat j} ight]$$
$$	herefore$$ $$\overrightarrow {OA}  + \overrightarrow {OB}  - \overrightarrow {OC} $$
$$ = \lambda \left[ {\left( {{{\sqrt 3  + 1} \over 2} + {1 \over {\sqrt 2 }}} ight)\widehat i + \left( {{1 \over 2} - {{\sqrt 3 } \over 2} - {1 \over {\sqrt 2 }}} ight)\widehat j} ight]$$
$$	herefore$$ Angle with x-axis
$${	an ^{ - 1}}\left[ {{{{1 \over 2} - {{\sqrt 3 } \over 2} - {1 \over {\sqrt 2 }}} \over {{{\sqrt 3 } \over 2} + {1 \over 2} + {1 \over {\sqrt 2 }}}}} ight] = {	an ^{ - 1}}\left[ {{{\sqrt 2  - \sqrt 6  - 2} \over {\sqrt 6  + \sqrt 2  + 2}}} ight]$$
$$ = {	an ^{ - 1}}\left[ {{{1 - \sqrt 3  - \sqrt 2 } \over {\sqrt 3  + 1 + \sqrt 2 }}} ight]$$
Hence option (a).

Answer

$${	an ^{ - 1}}{{(1 - \sqrt 3  - \sqrt 2 )} \over {(1 + \sqrt 3  + \sqrt 2 )}}$$

Answer

$${	an ^{ - 1}}{{(\sqrt 3  - 1 + \sqrt 2 )} \over {(1 + \sqrt 3  - \sqrt 2 )}}$$

Answer

$${	an ^{ - 1}}{{(\sqrt 3  - 1 + \sqrt 2 )} \over {(1 - \sqrt 3  + \sqrt 2 )}}$$

Answer

$${	an ^{ - 1}}{{(1 + \sqrt 3  - \sqrt 2 )} \over {(1 - \sqrt 3  - \sqrt 2 )}}$$

Motion in 2D & Vectors

The magnitude of vectors $\overrightarrow {OA} , \overrightarrow {OB} and \overrightarrow {OC} in the given figure are equal. The direction of \overrightarrow {OA} + \overrightarrow {OB} - \overrightarrow {OC} $ with x-axis will be :

Choose the correct answer: