 # State with reasons, whether the following algebraic operations

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