Introduction to Scalars and Vectors

• There are numerous examples of physical quantities – scalars and vectors.
• Some examples of physical quantities are time, the mass of an object, the weight of a liquid, the force applied to an object, and the electric current.
• Gravity weight plus the normal weight is considered the whole weight of an object. Running along the street, kicking a football, and experiencing electric charge through electricity are physical quantities in nature.
• These physical quantities are classified into two groups. They are

(i) Scalar quantity

(ii) Vector quantity

Scalar quantity and vector quantity

Scalar quantity

• Scalar quantity can tell you about the quantity of something or the size of some object. But it can’t tell whether the value is positive or negative, or it can’t tell you about the direction the object is traveling.
• It is one of the major differences between scalars and vectors. 
• Scalar quantity gives you a numerical value without any direction.
• For example, time and mass are examples of scalar quantities in which the numerical value and direction can’t be predicted.
• Some of the examples of scalars are 

1. length
2. area
3. volume
4. mass
5. speed
6. density
7. pressure
8. temperature
9. energy
10. entropy
11. work
12. power
13. angular frequency
14. number of moles

• In physics, the physical quantities that don’t change based on the coordinates of rotational translation are called scalar quantities.
• Displacement and distance tend to be similar in meaning, but both are different. One is a scalar quantity, and the other is a vector quantity. These both are considered as best examples of scalars and vectors. Displacement represents both the value (magnitude) and the direction, so it is a vector quantity. But distance represents only (magnitude), so it is a scalar quantity.

Vector quantity

• The physical quantity that shows both magnitude and direction is a vector quantity.
• The vector quantity in one dimension can be represented by the positive (+) or negative sign (-).
• It expresses the physical quantity in both values (magnitude) and direction.
• Vectors are represented graphically as an arrow. It connects two points.
• Otherwise, it is represented by a line segment.
• Examples of vector quantity are

1. displacement 
2. velocity
3. acceleration
4. momentum
5. force
6. lift
7. drag
8. weight
9. angular velocity
10. magnetic polarization
11. change in temperature
12. position

• One of the famous examples of a vector quantity is force because you can measure the quantity of force applied and the direction in which the force is applied.

Different types of vectors

Both scalars and vectors can represent the type of quality of any physical or mathematical object. There are eleven types of vectors. They are

• zero vector
• unit vector
• position vector
• co-initial vector
• like vector
• unlike vector
• coplanar vector
• collinear vector
• equal vector
• displacement vector
• negative vector

Zero vector or null vector

When the point of departure and the point of arrival of a vector coincide with each other, it is known as a zero vector or null vector. It doesn’t have any magnitude and direction.

For example

• Position vector at origin
• When the particle is at rest
• When the particles move in a uniform direction

Position vector

It determines the position of an object or the location of any point concerning its origin. The starting point of the position vector is at the origin and may end at any point.

For example

• The path taken by a car is the best example of a position vector. The car’s starting point is fixed, and it keeps on moving. As the car moves, the position of the vector will also change in direction, and the kilometers it travels will determine the magnitude.

Co-initial vector or Concurrent vector

Co-initial means when the point of origin is the same. Two or more vectors starting at the same point are called co-initial vectors. They may move in any direction, but these vectors have the same starting points.

For example

• XY and XZ are co-initial vectors because they have the same starting point, X.

Like vectors

Here the vectors move in the same direction.

For example

• If car A and car B travel in the east direction, which is in the same direction, they are like vectors.

Unlike vector

Here both vectors travel in opposite directions

For example

• Let us consider the above example of two cars. If both cars, A and B, move in the opposite directions, they are called unlike vectors.

Coplanar vectors

Here all the vectors lie in the same plane. To become coplanar, certain conditions should be followed.

For example

• Consider three or more vectors in the 3D plane, and their scalar product is zero; then, these vectors are called coplanar vectors.
• If three vectors in the 3D plane are linearly independent, these vectors are called coplanar vectors.

Collinear vectors or parallel vectors

Irrespective of magnitude and direction, the vectors should be parallel to the same line. The collinear vector should travel in the same line or parallel to each other in the same or opposite direction.

For example

• Parallel vectors can be scalar with another vector
• In 3D, the cross product of a collinear vector is zero

Equal vector

When two or more vectors travel in the same direction and have equal magnitude, they are equal vectors.

For example

• When two cars travel a distance of 20 kilometers in the west direction at a speed of 50 km/hrs, they are considered equal vectors as their magnitude and direction are the same.

Displacement vector

When vectors move from one place to another, they are called displacement vectors.

For example

• If a vector XY moves from position X to position Y, this is called a displacement vector.

Negative vector

Vectors that travel with the same magnitude but in opposite directions are called negative vectors.

For example

• Vector X has magnitude c and travels in a certain direction. Assume another vector Y has magnitude c but travels in the opposite direction, then vector X is the negative of vector Y. X= -Y has the same magnitude but different directions.

Scalar and vector examples

There are a number of examples for scalars and vectors. Here a few are listed out.

Scalar quantity examples

• Temperature of -100 degrees Celsius
• 100 kilocalories in the milk
• 200 km/h distance traveled by car
• 1024 megabytes

These are examples of a scalar quantity. Because the direction is not mentioned, sometimes a scalar quantity can be negative. Here -100 degrees doesn’t represent the direction. It represents the temperature.

The volume of the square to the north side of the apartment is 20 cubic feet.

Ans: It is scalar. You may think that the location is on the west side, but the location has nothing to do with the square. So here, only the volume of the square is considered (magnitude).

Vector quantity example

• 10 meters in the north direction
• 45 m/sec in the west direction

These two are examples of vectors as directions are mentioned

The baseball player walked 20 miles an hour towards the end zone

Ans: It is a vector quantity as 20 miles represent the magnitude and the directions towards the end.

The bike accelerated west at the rate of 10 meters per second square.

Ans: It is a vector quantity as acceleration comes under it.

Scalars and vectors difference

These are some of the differences between scalars and vectors.