Question

State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful:

1. adding any two scalars,
2. adding a scalar to a vector of the same dimensions,
3. multiplying any vector by any scalar,
4. multiplying any two scalars,
5. adding any two vectors,
6. adding a component of a vector to the same vector.

Answer 1

1. No, adding two scalars only makes sense when both represent the same physical quantity.
2. No, adding a scalar to a vector of the same dimension is not meaningful because a vector can only be added to a vector, and a scalar can only be added to a scalar.
3. Yes, multiplying any vector by any scalar is meaningful as multiplying a vector by scalar results in a new vector whose magnitude is equal to the product of the magnitudes of the vector and the scalar, and whose direction remains unchanged.
4. Yes, multiplying any two scalars is meaningful, as the magnitude of a new scalar obtained from the multiplication of two scalars is equal to the product of the magnitudes of the given scalars.
5. No, adding any two vectors is not meaningful because only vectors of the same dimensions (i.e., of the same nature) can be added.
6. Since a component of a vector is a vector that represents the same physical quantity as the original vector (for example, a component of force is also a force); therefore, adding a component of a vector to the same vector is meaningful.