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3-1.Vectors
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$\hat i\,\, + \;\,\hat j\,$ સાથેનો $3\hat i\,\, + \;\,4\hat j$ નો ઘટક ક્યો છે ?
A$\frac{1}{2}\,\,\left( {\hat j\,\, + \,\,\hat i} \right)$
B$\frac{3}{2}\,\,\left( {\hat j\,\, + \;\,\hat i} \right)$
C$\frac{5}{2}\,\,\left( {\hat j\,\, + \,\,\hat i} \right)$
D$\frac{7}{2}\,\,\left( {\hat j\,\, + \,\,\hat i} \right)$
Solution
ધારો કે $\mathop {\,a}\limits^ \to \,\, = \,\,3\hat i\,\, + \;\,4\hat j$ અને $\mathop {\,b}\limits^ \to \,\, = \,\,\hat i\,\, + \,\,\hat j\,$
$\mathop {\,b}\limits^ \to $ ને સમાંતર $\mathop {\,a}\limits^ \to $ નો ઘટક $,\,\overrightarrow b \,\,\left( {a\,\,\cos \,\,\theta } \right)\overrightarrow b \,\,\,$
$\frac{{\mathop {\,a}\limits^ \to \,.\,\,\mathop {\,b}\limits^ \to }}{{\rm{b}}}\,.\,\frac{{\mathop {\,b}\limits^ \to }}{{\mathop {\,b}\limits^ \to }}\,\, = \,\,\frac{{\left( {\mathop {\,a}\limits^ \to \,\,.\,\,\mathop {\,b}\limits^ \to } \right)\,\,\mathop {\,b}\limits^ \to }}{{{b^2}}}$
$ = \,\,\frac{{\left( {3\,\, + \;\,4} \right)\,\,\left( {\hat i\,\, + \,\,\hat j} \right)}}{2}\, = \,\,\frac{7}{2}\,\,\left( {\hat i\,\, + \,\,\hat j} \right)\,\,$
$\mathop {\,b}\limits^ \to $ ને સમાંતર $\mathop {\,a}\limits^ \to $ નો ઘટક $,\,\overrightarrow b \,\,\left( {a\,\,\cos \,\,\theta } \right)\overrightarrow b \,\,\,$
$\frac{{\mathop {\,a}\limits^ \to \,.\,\,\mathop {\,b}\limits^ \to }}{{\rm{b}}}\,.\,\frac{{\mathop {\,b}\limits^ \to }}{{\mathop {\,b}\limits^ \to }}\,\, = \,\,\frac{{\left( {\mathop {\,a}\limits^ \to \,\,.\,\,\mathop {\,b}\limits^ \to } \right)\,\,\mathop {\,b}\limits^ \to }}{{{b^2}}}$
$ = \,\,\frac{{\left( {3\,\, + \;\,4} \right)\,\,\left( {\hat i\,\, + \,\,\hat j} \right)}}{2}\, = \,\,\frac{7}{2}\,\,\left( {\hat i\,\, + \,\,\hat j} \right)\,\,$
Standard 11
Physics
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