Vector Algebra | Basic Physics for IIT JAM PDF Download

Introduction

• Vector is one of the most important for studying physics. But to know about vectors you should first know about a scalar. 
• A scalar is a quantity (length etc) which has only the magnitude (3 m or 4 m and so on) but it doesn't specify the direction as in 3 m to the south of a reference point on a reference plane. This is where vector comes in use as most of the quantity defined in physics involves direction along with the magnitude.
• Vector follows the rules of mathematics and geometry as well just in a different manner as the direction pokes its nose the basic concept. And while adding the magnitude of the vectors we are also supposed to keep in mind the direction.
•  In the following paragraph, you will learn to 'visualise' the concept as its he key aim of physics – to be able to 'see' things and not just mathematically or practically prove it.

Addition and subtraction of vectors

Before we dig deeper into the mathematical concepts, we first need to clarify that, vector deals only with the starting and the end points and not the path covered. There in the figure 1, there are several ways to go to point C from A via B, but the vector is meant to deal with the shortest path, that is the straight lines joining them.

Vector addition

If we want to add two vectors, there are two pairs of starting and ending points, and in order to add them, we need to move the second vector's starting point to the end point of the first vector without changing either of them's direction, and thus as a resultant we get two point in total which is the starting point of the first vector (which becomes starting point of the resultant vector) and the end point of the second vector (which becomes the end point of the resultant vector). And then the shortest path from the starting to the end point of the resultant vector gives the complete resultant vector. The concept is illustrated in the figure 2.

Similarly we can add three vectors and more subsequently as in Figure 3. 

Vector subtraction

Vector subtraction becomes a little tricky but uses the same concept as the addition, except that the direction of the subtrant is 'reversed' as in the starting and the end points of the second vectors interchange their positions and point in the opposite direction, and simultaneously, a minus (–) sign is added in front of the second vector. The concept is illustrated in Figure 4.

Vector multiplication

Vector multiplication is very different from scalar multiplication and it involves a further concept of resolving the vector into its components which is explained in the following section.

Resolving a vector into components
Resolving a vector into components is more understood visually than mathematically. Here we take an example of a vector that is lying at angle (theta)with respect to the horizontal reference. And we can picture the horizontal component of the vector if we place a light source in the vertical position looking down at the vector and the shadow that falls on the horizontal line is the required horizontal component of the vector. It's a scalar and it doesn't have a direction. Similarly the vertical component can be evaluated if we place the same light source on the left of the vector with its shadow falling on the vertical line. It is again a scalar quantity. The whole concept is illustrated in the Figure 5.

Now based on the previously discussed concept we can see that cos (theta) = X/A and sin (theta) = Y/A 
Therefore, the horizontal component X = Acos(theta) and horizontal component Y = Asin(theta)

Scalar (or 'dot') products

The dot product is a way of multiplying two vectors that results in a scalar (a single number). It's useful in physics for finding angles between vectors and for projections.
Definition

The dot product of two vectors A and B can be defined mathematically as:

A⋅B=∣A∣∣B∣cosθ

where:

• $\mid A\mid |\backslash mathbf\{A\}|$ is the magnitude (length) of vector A.
• $\mid B\mid |\backslash mathbf\{B\}|$ is the magnitude of vector B.
• $\theta \backslash theta$ is the angle between the two vectors.

Example:

Let's say you have two vectors:

• A = 3i + 4j (which represents a vector in 2D space)
• B = 2i + 1j

So, the dot product gives you a scalar value (10 in this case) that tells you something about the relationship between the two vectors. 

• If the dot product is positive, the angle between the vectors is less than 90°.
• If it's zero, the vectors are perpendicular.
• If it's negative, the angle is greater than 90°.

Note: Scalar product is commutative 

Vector (or 'cross') products

If A and B are two independent vectors, then the result of the cross product of these two vectors (Ax B) is perpendicular to both the vectors and normal to the plane that contains both the vectors. It is represented by:

A x B= |A| |B| sin θ

Cross Product of Two Vectors Meaning

a × b =|a| |b| sin θ.

Cross Product Formula

Cross Product of Two Vectors Properties

The properties of the cross product are essential for gaining a clear understanding of vector multiplication and simplifying various vector calculation problems. The characteristics of the cross product of two vectors include the following:

Triple Cross Product

The triple cross product involves taking the cross product of one vector with the cross product of the other two vectors. This results in a new vector, known as the triple cross product. This resultant vector lies in the plane defined by the three original vectors. If a, b, and c are the vectors, the vector triple product can be expressed as follows:

Examples 

1.  Twisting a bolt with a spanner: The length of the spanner is one vector. Here the direction we apply force on the spanner (to fasten or loosen the bolt) is another vector. The resultant twist direction is perpendicular to both vectors.

2.