question_answer

Which of the following quantities are independent of the choice of orientation of the coordinate axes : \[\vec{a}+\vec{b},3{{a}_{x}}+2{{b}_{y}},|\vec{a}+\vec{b}-\vec{c}|,\] angle between \[\vec{b}\] and \[\vec{c},\lambda \vec{a}\],where \[\lambda \] is a scalar?

Answer:

                A vector, its magnitude and the angle between two vectors do not depend on the choice of the orientation of the coordinate axes, so \[\vec{a}+\vec{b},|\vec{a}+\vec{b}-\vec{c}|,\]  angle between \[\vec{b}\] and \[\vec{c}\] and \[\lambda \vec{a}\]are independent of the orientation of the coordinate axes. But the quantity \[3{{a}_{x}}+2{{b}_{y}}\] depends upon the magnitude of the components along \[x\]- and y-axes, so it will change with the change in coordinate axes.