Why can’t you add a scalar to a vector?

Hint: Physical quantities can be categorized as Scalar and Vector depending on their magnitude and direction. Quantities that only have magnitude are called scalar. It is described using a number. But vectors have both magnitude and direction. A direction must be associated with the vector. Though adding a scalar to a vector is impossible since they have their different dimensions in space, it is possible to multiply a vector by a scalar.

Complete step by step solution:
A scalar quantity cannot be added to a vector quantity because they have different dimensions. A vector has both magnitude and direction but a scalar has magnitude only and has no direction. Addition of two scalar quantities is significant only if they both represent the same physical quantity. The addition of a vector quantity with a scalar quantity is not impossible. Scalar can be multiplied with a vector. For example, if force is multiplied with time to give impulse. Scalar, irrespective of the physical quantity it represents, could be multiplied with another scalar having the same or different dimensions, Addition of two vector quantities is significant only if they both represent the same physical quantity. The component of a vector can be added to the same vector as they both have the same kind of dimensions.

Note:When an object moves along a path joining two points the distance can be measured along the trajectory whereas the displacement is the shortest path joining the two points. The distance varies if the object follows the trajectories between the initial and final positions. The displacement of an object is independent of the path followed by the object. Hence we can say that distance is a scalar quantity but displacement is a vector quantity.