Table of Contents

**Introduction**

Physics is a simple mathematical science. There is a mathematical foundation for fundamental notions and principles. Throughout our physics studies, we will come across a variety of topics that have a mathematical foundation. While we will place a strong focus on the conceptual nature of physics, we will also pay close attention to its mathematical aspect. Physics makes use of a wide range of terms to describe motion.

Two types of mathematical quantities are used to explain the motion of objects. The value can be a vector or a scalar. The distinction between these two categories can be seen in their definitions. This article will cover both of the aforementioned categories.

**Overview**

The continuation of this module will cover a variety of vector and scalar examples (distance, displacement, speed, velocity, and acceleration). Pay close attention to the vector and scalar nature of each quantity as you progress through the course. We will also cover dot product or scalar multiplication, vector multiplication, & matrix representation.

Physical quantities can be represented using vectors. Vectors are extensively used in physics to express displacement, velocity, and acceleration. Vectors are arrows that represent a combination of magnitude and direction. The magnitude is represented by the length, and the direction of that amount is the vector’s pointing direction. Physical quantities (with both size and direction) can be analyzed as vectors because vectors are produced in this way.

**What are the Scalar Quantities? **

Scalar quantities are quantities that can be defined entirely by a single numerical value (or magnitude).

Real numbers, which are usually but not always positive, are used to characterize scalars. When a force acts on a particle, the work done by the force is a negative quantity when the particle moves in the opposite direction. Ordinary algebraic laws can be used to manipulate scalars.

For example Distance, Mass, Energy, etc.

Get the most Important Questions of Physics, Maths and Biology

**What are Vector Quantities?**

A vector is a quantity that is completely specified in its magnitude and direction. A quantity with magnitude and direction must also follow specific combination criteria to qualify as a vector.

Vector addition, expressed symbolically as A + B = C, is one of them.

If A, B, and C are vectors, the same operation must be doable in reverse order to achieve the same outcome (C), B + A = C. This is a property of quantities like displacement and velocity.

For example Velocity, Force, Displacement, etc.

**The product of two vectors is called the dot product or the scalar product**

Dot product of two vectors ( also known as scalar product) of two vectors a and b is defined as

ab =abcos?

where a & b represents magnitudes of a & b respectively, and ? is the angle between them.

The dot product between the mutually perpendicular vectors is always zero as cos90°=0

The Dot product follows commutative and distributive law

ab=ba

**Two vectors combined into a cross product or vector**

The process of multiplying two vectors is called the cross product or the vector product. A cross-product between two vectors is indicated by the multiplication sign(x). It’s a three-dimensional system with a binary vector operation. The third vector is the cross product of the two initial vectors and is perpendicular to the first two.

The magnitude of this vector is:

ab=absin?

where ? is the smaller angle between the two, a and b are the magnitudes of a & b respectively.

**Properties of the cross product of vector**

The cross-product features are useful for properly understanding vector multiplication and for quickly solving all vector calculation issues. Two vectors are cross-products when they have the following properties:

It does not follow commutative property abba

It follows Distributive property a(b+c)=(ab)+(ac)

Cross product with itself is always zero

**Matrix Representation**

Column vectors are matrices with a single column, and row vectors are matrices with a single row. Lower case letters printed in boldface are commonly used to depict vectors.

**Importance of this chapter from JEE and NEET point of view**

Vectors are used in many disciplines of physics where quantities must be characterized by both direction and magnitude. Displacement, velocity, acceleration, force, momentum, lift, thrust, friction, and weight are all vector values (aerodynamic forces).

As an example of how vectors are used, the temperature of a particular medium is measured as a scalar quantity, but it is measured as a vector quantity when the temperature of the medium decreases or increases.

Electromagnetic laws and Maxwell’s equations are expressed using vectors and vector field concepts.

**Frequently Asked Questions**

**Question 1:** What’s the relation between vector and scalar products?

**Solution:** It is possible to multiply a vector by another vector, but not to split it. In physics and engineering, there are two types of vector products that are commonly employed. One sort of multiplication is scalar multiplication of two vectors. The scalar product of two vectors produces a number, as the name implies (a scalar).

Scalar products are used to define the link between energy and work. For example, the work that a force (a vector) does on an object while generating its displacement is defined by the scalar product of the force vector and the displacement vector (a vector).

A vector multiplication of vectors is a sort of multiplication that is quite different. As the name implies, the vector product of two vectors produces a vector. Vector products are used to define other derived vector values.

In defining rotations, a vector variable called torque is defined as the vector product of an applied force (a vector) and its distance from the pivot to the force (a vector). It is critical to distinguish between the two types of vector multiplications because a scalar product is a scalar quantity and a vector product is a vector quantity.

**Question 2:** Recognize the differences between a scalar and a vector quantity.

**Solution 2: **

Scalar | Vector |

A scalar quantity requires only magnitude, there is no need for direction.
A Scalar quantity is always 1 Dimensional It changes with a change in its magnitude. |
Vector quantities require magnitude as well as direction. A Vector quantity can be 2 or 3 Dimensional. It changes with change in its magnitude or change in its direction or from both. |

**Question 3:** How are vectors used in real life?

**Solution 3:** Vectors are useful in a variety of scenarios, including those involving force or velocity. Consider the pressures operating on a boat as it crosses a river. Motors create forces in one direction, while river currents create forces in the opposite direction. Vectors represent these forces.