# State with reasons, whether the following algebraic operations

State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful
(i) Adding any two scalars.
(ii) Adding a scalar to a vector of the same dimensions.
(iii) Multiplying any vector by any scalar.
(iv) Multiplying any two scalars.
(v) Adding any two vectors.
(vi) Adding a component of a vector to the same vector.

(v) Adding any two vectors. "
(vi) Adding a component of a vector to the same
vector. [NCERT]
Sol. (i) No, adding any two scalars is not meaningful because only the scalars of same dimensions i.e. having same unit can be added. [1/2]
(it) No, adding a scalar to a vector of the same dimensions is not meaningful because a scalar cannot be added to a vector. [1/2]
(iit) Yes, multiplying any vector by any scalar is meaningful. When a vector is muldplied by a scalar we get a vector, whose magnitude is equal to the product of magnitude of vector and the scalar and direction remains the same as the direction of the given vector.
e.g., A body of mass 4 kg is moving with a velocity 20 m/s towards East then, product of velocity and mass gives the momentum of the body which is also a vector quantity.
p = mv = 4 kg x (20 m/s) (East)
= 80kg-m/s, East
(iv) Yes, multiplying any two scalars is meaningful.Density p and volume V both are scalar quantities. When density is multiplied by volume, them we get p X V = m, mass of the body, which is a scalar quantity.
(v) No, adding any two vectors is not meaningful because
only vectors of same dimensions i.e. having same unit can be added.
(vi) Yes, adding a component of a vector to the same vector
is meaningful because both vectors are of same dimensions.