Scalars and Vectors - Definition, Examples, Types, FAQs

Physical Quantity:

A physical quantity is a matter or system property that can be measured and quantified. At least two attributes are held by all physical quantities. The first is the numerical magnitude, and the second is the unit of measurement.

Types of physical quantities:

There are seven basic physical quantities.

Length, Mass, Time, Temperature, Current, luminous intensity, and amount of substances.

Some quantities in physics have simply magnitude, some have both magnitude and direction. It is fundamental to know the properties of vectors and scalars in order to understand these physical quantities.

What is Scalar quantity and give scalar quantity examples:

Scalar definition and Scalar meaning: A scalar property is one that can only be described in terms of magnitude. Scalars can be used to describe a variety of quantities in physics. Distance, mass, temperature, speed, and energy are only a few examples. A scalar value is a single value; a scalar number can never be negative because it just has magnitude and no direction.

For instance, consider the terms string number, variable, and column. In contrast to a set of values, a scalar value is a single value. Every point in space is represented mathematically as a scalar value.

A scalar is a field element that is used to define a vector space. Physical scalar fields can be thought of as a subset of more general fields like vector fields, spinor fields, and tensor fields, while scalars are usually thought of as special cases of multi-dimensional quantities like vectors and tensors.

Relativistic scalar: The variations between the outcomes of (four-component, two-component, or one-component “scalar”) relativistic quantum theoretical computations using a finite and infinite speed of light are known as relativistic effects. Electric charge, proper time and proper length space time intervals, and invariant mass are all scalar quantities in relativity.

The Higgs field is the only fundamental scalar quantum field that has been detected in nature. Scalar quantum fields, on the other hand, appear in many effective field theory models of physical processes.

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What is vector quantity and give vector quantity examples:

Vector definition and Vector quantity meaning: Vector is a quantity that can be described in both the terms of magnitude and direction. A vector is a directed line segment in geometry. Certain quantities in physics can only be described using vectors.

Force, velocity, displacement, position vector, acceleration, linear momentum, and rotational momentum are all examples.

Magnitude of a vector:

The length of a vector determines its magnitude. The magnitude of a vector is sometimes referred to as the vector's norm. For a vector A, the magnitude or norm is denoted by A.

Different types of vector:

• Equal vectors

a) Collinear vectors

i) Parallel vectors

ii) Anti-parallel vectors

• Unit vector
• Orthogonal unit vectors

Equal vectors:

Two vectors A and B are said to be equal since it have equal magnitude as well as same direction along with represent the same physical quantity.

a) Collinear vectors:

Vectors that behave in the same direction are known as collinear vectors. The angle between them can be 0oor 180o

i) Parallel vectors:

The angle formed by two vectors A and B acting in same direction along same line is 0o.

ii) Anti-parallel vectors:

When two vectors A and B are in opposite directions along same line, they are said to be antiparallel. The angle between them is then 180o.

Unit vector:

A unit vector is a vector whose magnitude is divided by its length. The unit vector for A is denoted by A. It has a magnitude of one or unity. As a result, we can declare that the unit vector just provides the vector quantity's direction.

Orthogonal unit vectors:

i, j and k are examples of orthogonal vectors. Orthogonal vectors are two vectors that are perpendicular to each other.

Addition of vectors:

The approach of standard algebra cannot be used to add vectors since they have both magnitude and direction. As a result, vectors can be added geometrically or analytically using vector algebra methods. The triangular law of addition or parallelogram law of vectors approach is used to discover the sum or resultant of two vectors that are inclined to each other.

By triangular law of addition method,

Magnitude of resultant vector,

The direction of resultant vectors,

Subtraction of vectors:

Because vectors have both magnitude and direction, the approach of regular algebra cannot be used to subtract two vectors. As a result, this subtraction can be done analytically or geometrically.

The magnitude of the resultant vector,

The direction of resultant vectors

Multiplication of vector by a scalar:

A vector A multiplied by a scalar results in another vector A. If is a positive number, then A is also in the direction of A. It is a negative number, then A is in the opposite direction to the vector A

Scalar product of two vectors:

Definition:

The scalar product, often known as the dot product of two vectors, is made up of vectors and the angle's cosine.

Thus, if there are two vectors A and B having an angle of between them, then their scalar product is defined as A . B=AB cosθ. Here, A and B are magnitudes of A and B.

Properties of scalar product:

Note: Here A, B and C are vectors.

• The product quantity A . B is always a scalar. It is positive, when the angle between the vectors is acute i.e. <90o and it is negative if the angle between them is obtuse i.e. 90^o<<180^o
• The scalar product is commutative i.e. A . B=B . A
• The vectors obey distributive law i.e. A . (B+C)=A . B+A.C
• The angle between the vectors =A.B/|AB|
• The scalar product of two vectors will be maximum when cosθ=1 i.e. =0^o, when the vectors are parallel, (A . B)_max=AB
• The scalar product of two vectors will be minimum, when cosθ=-1 i.e. =180^o, when the vectors are antiparallel, (A . B)_min=-AB
• The scalar product of a vector with itself is termed a self-dot product.
• If two vectors A and B are perpendicular to each other, then their scalar product is A.B=0, because cos90^o=0. Then the vectors are said to be mutually orthogonal.

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Vector product of two vectors:

Definition:

The vector product is a vector with a magnitude equal to the product of the magnitudes of two vectors plus the sine of the angle between them, also known as the cross product of two vectors. The product vector is perpendicular to the plane containing the two vectors, according to the right-hand screw rule or right-hand thumb rule.

Properties of vector product:

Note: Here A, B and C are vectors.

• It is not commutative to take the vector product of two vectors. A × B=-[B ×A]
• The magnitude of the vector product of two vectors will be the greatest, when sinθ=1 that is =90^o, When the vectors A and B are perpendicular to one another.

(A × B)_max=

• The vector product of the two non-zero vectors will be minimum when sinθ=0 i.e. =0oor 180^o

(A × B)_min=0

i.e. If the vectors are parallel or antiparallel, the vector product of two non-zero vectors vanishes.

• The null vector is the self-cross product, which is the product of a vector with itself. In physics, the null vector is simply referred to as zero.

Vector products are used to define a variety of quantities in physics. The vector products are used to construct physical quantities that represent rotational effects, such as torque and angular momentum.

Position vector:

A position vector is a scalar that represents the position of a particle in relation to some reference frame or coordinate system at any given time.

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NCERT Physics Notes:

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