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.