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Let $\vec A\, = \,(\hat i\, + \,\hat j)\,$ and $\vec B\, = \,(2\hat i\, - \,\hat j)\,.$ The magnitude of a coplanar vector $\vec C$ such that $\vec A\cdot \vec C\, = \,\vec B\cdot \vec C\, = \vec A\cdot \vec B$ is given by
$\sqrt {\frac{5}{9}} $
$\sqrt {\frac{10}{9}} $
$\sqrt {\frac{20}{9}} $
$\sqrt {\frac{9}{12}} $
Solution
$\begin{array}{l}
If\,\vec C = a\hat i + b\hat j\,then\,\vec A.\vec C = \vec A.\vec B\\
a + b = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,…\left( i \right)\\
\vec B.\vec C = \vec A.\vec B\\
2a – b = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,…\left( {ii} \right)\\
Solving\,equation\,\left( i \right)\,and\,\left( {ii} \right)\,\,we\,get\\
a = \frac{1}{3},\,b = \frac{2}{3}\\
Magnitude\,of\,coplanar\,vector,\\
\left| {\vec C} \right| = \sqrt {\frac{1}{9} + \frac{4}{9}} = \sqrt {\frac{5}{9}}
\end{array}$
Similar Questions
Vector $A$ is pointing eastwards and vector $B$ northwards. Then, match the following two columns.
| Colum $I$ | Colum $II$ |
| $(A)$ $(A+B)$ | $(p)$ North-east |
| $(B)$ $(A-B)$ | $(q)$ Vertically upwards |
| $(C)$ $(A \times B)$ | $(r)$ Vertically downwards |
| $(D)$ $(A \times B) \times(A \times B)$ | $(s)$ None |
