Scalar, Vector Quantities and Units

Dr. Cameron Ahmad
3 min readApr 4, 2021

The units at present prevalent across the scientific world can be expressed under the MKS convention meaning Length in Meters (m or cm), Mass in Kg or gm, and finally Time in Seconds. Else they can be written under the FPS convention, meaning Length is in Feet (ft), Mass is in Pounds (lbs) and Time is in Seconds. So, let us check out one by one based on the convention (also called system) to write the units of physical quantities.

However, the quantities we encounter in our daily life can be from one of the two types. First are Scalar quantities that have only “magnitude” such as Mass, Energy, Speed, Temperature, Volume, Area, Length, Density, etc. The second type is grouped under the Vector quantities. They have both a magnitude and direction. It sounds very odd to digest if a quantity will have both magnitude and direction. So, let us do a thought experiment.

Say, a child wants to pull out candies kept by parents inside a cabinet, where his/her hand does not reach due to his/her short height. However, if the child stands on a chair then he/she can reach the cabinet and bring out the candies. Here we have the assumption the child has enough strength to drag a chair and then climb on it to reach the cabinet. So as far as strength is considered it is a scalar quantity and did not require any direction but only magnitude.

But when the child applies this strength to push the chair to reach below the cabinet, it means the chair was made to move from its existing position of rest. Kindly recall here the 1st law of motion viz., a body (in this case the chair) continued in a state of rest (or uniform motion) unless it had been acted upon by an external force (applied by the child in this case). So, the child applies his strength to make the chair move in the direction of the cabinet and reach near it, then stops applying his strength to move it any further. In other words, the applied force by the child ceased to remain acting once the chair reached the required location and returns to a position of rest. But had the child not applied the strength in the direction he/she wanted the chair to move, could the chair have moved despite the child had all the strength to move it. The answer is NO! It implies the magnitude of strength in combination with the direction in which it was applied is nothing but the Applied Force by the child on the chair to move.

Hence, we see Force as a Vector quantity while strength is a scalar quantity. Examples of other Vector quantities are Displacement, Velocity, Acceleration, Momentum, Field, Temperature Increase/Decrease, etc.
Let us return to the road once again to understand a scalar quantity by the signs displayed representing the “Speed Limit”! Such signage never writes it as the Velocity Limit with Direction, because the vehicle is already moving in a fixed direction when it encounters the signage about the maximum rate at which the vehicle can move that is allowed in the direction it is moving on the road. On the signage, no direction is ever mentioned!
Now to understand and determine the units, it can be arrived at by the sheer definition as an example of say three physical quantities in the following manner.

The magnitude of scalar quantities such as Length, Area, Volume, Temperature has their units as Meters, Sq Meters, Cubic Meters, Centigrade, etc respectively.

Velocity: it is the distance per unit time in a particular direction (say North, South, East, West, South East, etc.). In the MKS system length is given by Meters (m) and Time by Second (s). So, the units for the velocity will be given as m/s, along with the direction as required.

Acceleration is defined as the Rate of Change of Velocity along with a direction. Hence, the units for it will be Velocity/Time. Or we can write it as m/s/s (meters per second squared). Which is nothing but m/sec² and a direction.

Force is defined as mass multiplied by Acceleration viz., F = Mass * Acceleration. In MKS the Mass is given by Kg, so the unit of force will then become Kg * m/sec². This is termed Newton meaning 1 Newton amount of Force is being exerted when a mass of 1 Kg is moved with an acceleration of 1 m/sec² in a particular direction.

Cameron Ahmad

Code: PHSK-5

April 03, 2021

Brampton ON

Canada

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Dr. Cameron Ahmad

B. Sc (Hon) in Physics, M. Sc (Biophysics & Electronics). M. Tech (Applied Optics), PhD (Engineering Science), PMP, RDCS, DMS, CET, AScT, CTDP & CECC