Motion in a Plane

Scalars and Vectors

In today’s world, various mathematical quantities depict the motion of objects into two categories. The quantity is as either a vector or a scalar quantity which distinguishes from one another by their difference and distinct definitions. Let’s study more about the scalars and vectors below.

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Scalar and vector quantities
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Scalars and Vector Quantities

  • Scalar Quantities: The physical quantities which are specified with the magnitude or size alone are scalar quantities. For example,  length, speed, work, mass, density, etc.
  • Vector Quantities: Vector quantities refer to the physical quantities characterized by the presence of both magnitude as well as direction. For example, displacement, force, torque, momentum, acceleration, velocity, etc.

Vectors

Comparison between Scalars and Vectors

Criteria Scalar Vector
Definition A scalar is a quantity with magnitude only. A vector is a quantity with the magnitude as well as direction.
Direction No direction Yes there is the direction
Specified by A number (Magnitude) and a Unit A number (magnitude), direction and a unit.
Represented by Quantity symbol Quantity symbol in bold or an arrow sign above
Example Mass and Temperature Velocity and Acceleration

Characteristics of Vectors

The characteristics of vectors are as followed –

  • They possess both magnitudes as well as direction.
  • They do not obey the ordinary laws of Algebra.
  • These change if either the magnitude or direction change or both change.

Unit Vector

A unit vector is that vector which is a vector of unit magnitude and points in a particular direction. The unit vector in the direction of  \( \vec{A} \) is \( \hat{A} \) and is defined by –

| A | \( \hat{A} \) = \( \vec{A} \)

The unit vectors along the x, y and z-axis is  \( \hat{i} \), \( \hat{j} \), and  \( \hat{k} \) respectively.

Equal Vectors

Vectors A and B are equal if | A | = | B | as well as their directions, are same.

Zero Vectors

Zero vector is a vector with zero magnitudes and an arbitrary direction is a zero vector. It can be represented by O and is a  Null Vector.

Vectors

Negative of a Vector

The vector whose magnitude is same as that of a (vector) but the direction is opposite to that of a ( vector ) is referred to as the negative of a ( vector ) and is written as – a ( vector ).

Parallel Vectors

A and B are said to be parallel vectors if they have the same direction, or may or may not have equal magnitude ( A || B ). If the directions are opposite, then A ( vector ) is anti-parallel to B ( vector ).

Coplanar Vectors

If the vectors lie in the same plane or they are parallel to the same plane, the vectors are said to be coplanar. If not, the vectors are said to be non – planar vectors.

Displacement Vectors

The displacement vector refers to that vector which gives the position of a point with reference to a point other than the origin of the coordinate system.

Solved Question for You

Question 1: State for each of the following physical quantities, if it is a scalar or a vector.

Volume, Mass, Speed, Velocity, Displacement, Acceleration, Density, Number of Moles, Angular Frequency, Angular Velocity, Displacement

Solution.

  • A scalar is the one that is specified by its magnitude. It does not have any direction associated with it. Some of the scalar physical quantities are – Volume, Speed, Mass, Density, Number of Moles and Angular Frequency.
  • Whereas a Vector quantity is the one which is specified by its magnitude as well as its direction that is associated with it. Some of the vector quantity are velocity, acceleration, displacement and angular velocity.

Therefore,

  • Scalar: Volume, Mass, Speed, Velocity, Density, Number of Moles, Angular Frequency
  • Vector: Acceleration, Velocity, Displacement, Angular Velocity.

Question 2: Pick out the two scalar quantities in the list.

Force, Work, Angular Momentum, Current, Linear Momentum, Electric field, Average Velocity, Relative Velocity, Magnetic Moment

Solution. The scalar quantities from the mentioned are Work and Current.

  • Work done is the dot product of force and displacement. Since the dot product of two quantities is always a scalar, therefore, work is a scalar physical quantity.
  • Current refers to the one in which the direction is not taken into consideration and is described only by its magnitude. Henceforth, it is a scalar quantity.
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