कारण सहित बताइए कि अदिश तथा सदिश राशियों के साथ क्या निम्नलिखित बीजगणितीय संक्रियाएँ अर्थपूर्ण हैं ?

$(a)$ दो अदिशों को जोड़ना,

$(b)$ एक ही विमाओं के एक सदिश व एक अदिश को जोड़ना,

$(c)$ एक सदिश को एक अदिश से गुणा करना,

$(d)$ दो अदिशों का गुणन,

$(e)$ दो सदिशों को जोड़ना,

$(f)$ एक सदिश के घटक को उसी सदिश से जोड़ना

$(a)$ Meaningful : The addition of two scalar quantities is meaningful only if they both represent the same physical quantity. 

$(b)$ Not Meaningful : The addition of a vector quantity with a scalar quantity is not meaningful.

$(c)$ Meaningful : A scalar can be multiplied with a vector. For example, force is multiplied with time to give impulse.

$(d)$ Meaningful : A scalar, irrespective of the physical quantity it represents, can be multiplied with another scalar having the same or different dimensions.

$(e)$ Meaningful : The addition of two vector quantities is meaningful only if they both represent the same physical quantity.

$(f)$ Meaningful : A component of a vector can be added to the same vector as they both have the same dimensions.