Vectors and Scalars

Contents of the Module

* Position and displacement vectors * Equality of vectors * Vector Addition and subtraction * zero vector * scalar and vector products * Unit Vector * Resolution of a Vector.

Learning outcomes of this Module

1.Understanding Scalar and Vector Quantities: * Learning Outcome: Students will be able to distinguish between scalar and vector quantities, identifying examples of each in real-world scenarios. 2.Applying Elementary Concepts of Differentiation and Integration: *Learning Outcome: Students will gain proficiency in applying basic concepts of differentiation and integration to describe motion, enabling them to analyze velocity and displacement in various scenarios. 3.Manipulating Position and Displacement Vectors: * Learning Outcome: Learners will develop the ability to work with position and displacement vectors, including calculating displacement vectors between two points and understanding their geometric representations. 4.Understanding Equality of Vectors: * Learning Outcome: Students will be able to recognize and apply the concept of vector equality, crucial for problem-solving in physics and engineering applications. 5.Performing Vector Addition and Subtraction: * Learning Outcome: Learners will gain proficiency in adding and subtracting vectors, providing them with problem-solving skills for scenarios involving forces, velocities, and displacements. 6.Analyzing the Zero Vector: * Learning Outcome: Students will understand the significance of the zero vector in various contexts, recognizing its role in denoting absence or equilibrium of certain physical quantities. 7.Applying Scalar and Vector Products: * Learning Outcome: Learners will be able to apply scalar and vector products in practical situations, such as calculating work done, determining torque, and understanding their applications in physics. 8.why dot product is scalar? 9. why only cos theta in dot product? 10.what happens if cross product is used in calculation of work instead of dot product? 11.Utilizing Unit Vectors: * Learning Outcome: Students will be proficient in using unit vectors to represent directions in 3D space, applying them in computer graphics and other relevant contexts. 12.Resolving Vectors: *Learning Outcome: Learners will develop the skill to resolve vectors into components, especially in scenarios involving inclined planes, aiding in the analysis of forces and motion.

Introduction

Step into the world of Vrindavan, where Little Krishna teaches Arjuna beneath the ancient Kadamba tree. Explore the exciting journey of learning and laughter.

Little Arjuna, practicing arrow shooting alongside Krishna, observed the wind shifting his arrows. Curious, he asked Krishna, 'Why does the wind guide them?' With a smile, Krishna explained, 'The wind is like a friend, sometimes changing the direction to the left, sometimes to the right, and at times increasing or decreasing the speed of your arrows."

Arjuna, still curious, remarked, 'So everything in nature has force and a direction?' Krishna, with a wise smile, corrected him, 'Not all. Observe the sunlight, Arjuna. It has only influence and no specific direction; it warms everything it touches. Nature is diverse, and each element plays its unique role. Not everything follows the same rules of force and direction.

Arjuna remarked, 'So, there are things in nature with influence but no set direction. I understand now, but how do we know which ones have influence and a direction and which ones have only influence?' Krishna,

Nurturing Arjuna's curiosity, Krishna explained, 'Observation, Arjuna. The more we observe and learn about nature, the better we understand its patterns.'

'Some elements, like the wind, exhibit both force and direction.'

'Others, like sunlight, display influence without a specific path.'

Guided by Krishna's wisdom, Arjuna improved his archery, hitting the target perfectly every time.

As we bid farewell to the magical tales of Arjuna and Krishna in Vrindavan, we take with us a lesson about nature's language. Unveiling two fundamental aspects—vectors and scalars—that weave the fabric of the universe, we're ready to explore how things move and work in the exciting journey ahead.

Vectors are like the playful wind guiding Arjuna's arrows. They can change both how much something happens(magnitude) and which way it goes(Direction).

On the other hand, scalars are more like the gentle warmth from sunlight. Just as sunlight warms things without changing its path, scalars only affect how big something is(Magnitude), not which way it goes(Direction).

Definitions of vectors and scalars:

Vectors:

A vector is a quantity that has both magnitude (size or amount) and direction. In other words, it represents a physical quantity that has a specific size and is associated with a particular direction.

Scalars:

A scalar is a quantity that has only magnitude (size or amount) and no specific direction. Scalars are used to represent physical quantities that can be described by their size alone, without indicating any particular direction.

In summary, vectors have both size and direction, while scalars have only size.

Direction of vectors:

Vectors, dealing with both size and direction, show up differently in various situations. Let's see how they work in one dimension (like a straight line), two dimensions (like a flat surface), and three dimensions (like moving all around us in every direction).

1-Dimention(1-D) Vectors:

A 1-dimensional vector is a quantity that has both magnitude and direction along a single straight line.

For example a car moving only along a straight road. The speed at which it moves represents the magnitude, and the direction it travels (either forward or backward) gives it a sense of direction.

For instance, if a car moves northward at 50 km/h, we express its velocity as 50 km/h in the j direction. The j direction signifies North-South, pointing northward with a magnitude of 50 km/h. Conversely, if the car moves southward at 50 km/h, we represent this as −50 km/h in the j direction, with the negative sign indicating southward movement.
An elevator ascending at a speed of 20 m/s. We express its velocity as 20 m/s20m/s in the k direction, where the k direction denotes vertical or upward movement. Conversely, if the elevator descends at 15 m/s, we represent this as −15 m/s in the k direction, with the negative sign signifying downward movement.

2-Dimensional (2-D) Vectors:

A 2-dimensional vector is a quantity that has both size and direction and operates within a plane.

For example, think of a bird moving east at a speed of 20 km/h. At the same time, if it's going up with a speed of 10 km/h, we express this movement as velocity=20 km/h i + 10 km/h k.

3-Dimensional (3-D) Vectors:

A 3-dimensional vector is a quantity that possesses both magnitude and direction in three different directions—horizontally (i), vertically (j), and in-depth (k).

For example, an object, like a spaceship, moving in three dimensions. If it's moving eastward (i) with 200 km/h, northward (j) with 150 km/h, and upward (k) with 100 km/h, we can represent its velocity as follows:

Velocity = 200 km/h i + 150 km/h j + 100 km/h k.

Rules for combining of scalars:

scalars can be combined through simple arithmetic operations. Here are the basic rules for combining scalars:

Addition: Scalars in physics can be added together to find the total quantity. For instance, if you have a distance of 5 m and another distance of 3 m, the total distance (D total) is given by:

D total=5m+3m=8m.

Subtraction: Subtracting scalars in physics is used to find the difference between two quantities. If an object undergoes a temperature change of 8 °C and then experiences a further change of 3 °C, the net change (ΔT) is given by:

ΔT=8°C−3°C=5°C.

Multiplication: Scalars can be multiplied to scale a quantity. For example, if you have a speed of 4 m/s and you travel for 2 s, the total distance (D total) covered is:

D total=4m/s×2s=8m.

Division: Division is used when you need to find a rate or ratio of two scalar quantities. If an object undergoes a volume change of 12 cm3in 4s, the average rate of change (Rate) is calculated as:

Rate=12cm3/4s=3cm3/s.

These basic arithmetic operations make it convenient to manipulate and calculate with scalar quantities in physics.