Education

# Understanding Scalars and Vectors from Scratch

In physics, we use some terms as scalar while others as vectors. Physics terms such as work, speed, force etc., are classified as scalar or vector quantities. Both types of quantities represent the motion of an object. In this article, we will know everything about scalars and vectors and the difference between Scalar and Vector in detail.

## What is Scalar?

A quantity that does not depend on direction is called a scalar quantity. A scalar quantity has only magnitude (or numerical value) alone. Scalar quantities can be written just by their numerical value without directions. For example, speed, energy, volume, mass, density, time etc.

With more specifications, 11 m and 15 ms-1 are both scalar quantities. Scalar quantity’s measurement does not change with the direction of action. For scalars, you only have to compare the magnitude. Scalar quantity changes with change in magnitude.

## What is Vector?

A vector is an object that has both a magnitude and a direction. For example, force, velocity, momentum, weight, electromagnetic fields etc., have specific directions and magnitude. With more specifications, both vector quantities are 11 m east and 15 ms-1 at 30⁰ to the horizontal.

Vector is very helpful in the study of motion. It is represented by an arrow whose direction is the same as that of the quantity. Its length is proportional to the magnitude. Vector quantity changes when their magnitude changes, their direction changes or their magnitude and direction both change.

A vector whose magnitude is 1 and direction is known as Unit Vector and can be written as a lower case alphabet with a hat on it. Like ŷ for vector y if y is a unit vector.

### Examples

Distance, mass, energy, volume, area, length, time, direct current (DC), frequency, speed, amount of substance, luminous intensity, density, concentration, refractive index, pressure, power, charge, electric potential, entropy, work, temperature, speed etc. are examples of scalar quantities.

While polarisation, acceleration, impulse, velocity, angle, alternating current, the torque of couple, weight, thrust, electric field strength, magnetic field strength, gravitation, frictional force, tension, momentum, force, electric field, displacement, etc. are examples of vector quantities.

With more specifications, speed is a scalar quantity – it is the rate of change in the distance travelled by an object, while velocity is a vector quantity – it is the speed of an object in a particular direction.

## Addition and Subtraction of Vectors

Vectors can not be added arithmetically. They need to be added geometrically. A special set of rules are followed for the addition and subtraction of vectors. Vector addition is finding the resultant of a number of vectors acting on a body.

The result is order independent, so the order of operation won’t change the answer. Each vector acts as if it is the only vector, so results should be calculated independently.

Vector subtraction is the same as vector addition. In short, vector subtraction = final vector – initial vector while scalar subtraction = large value – small value.

### Multiplication of Vectors

Vector multiplication is the multiplication of vectors by dot product, cross product, Hadamard product, outer product, triple product or multiple cross products.

### Similarities between Scalar and Vector

The similarities between scalar and vector are as below:

• Both scalar quantity and vector quantity is used to express physical entity
• Both can be represented using numbers because both are measurable
• Both have a certain finite magnitude
• Both have a specific dimension
• Both have a specific unit
• Both are measurable quantities

### Difference between Scalar and Vector

The difference between scalar and vector are as below:

• A scalar quantity has only magnitude and no direction. While vector quantity has both magnitude and direction
• Scalar is one dimensional while vector is two dimensional or three dimensional
• Scalar quantities divide each other while vector quantities do not divide each other
• Scalar is used to solve one-dimensional problems, while vector is used to solve multi-dimensional problems
• Scalar quantity can be changed by changing its magnitude. While vector quantity can be changed with a change in magnitude and direction
• Normal operational rules of addition, subtraction, multiplication and division can be applied to scalar quantities. Rules of operations like addition, subtraction, multiplication are different for vector quantities than arithmetic operational rules.
• You can divide one scalar quantity with another while not dividing one vector quantity with another.
• Scalar quantity can be specified by a number and a unit, while vector quantity can be specified by a number, direction and a unit.
• A quantity symbol represents a scalar quantity, while vector quantity is represented by a quantity symbol in bold or an arrow sign above.

## Conclusion

In this article, we have seen what a scalar quantity and a vector quantity is. We also discussed similarities and differences between them with a few examples each. We hope it helped you to understand the topic and get an overall idea about the same.

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