Vector Algebra | Basic Physics for IIT JAM PDF Download

Vectors

Introduction

Vector quantities have both direction as well as magnitude such as velocity, acceleration, force and momentum etc. We will use  for any general vector and its magnitude by ||. In diagrams vectors are denoted by arrows: the length of the arrow is proportional to the magnitude of the vector, and the arrowhead indicates its direction. Minus  is a vector with the same magnitude as  but of opposite direction.

Vector Operations

We define four vector operations: addition and three kinds of multiplication. 

(i) Addition of two vectors

 Place the tail of  at the head of ; the sum,, is the vector from the tail of  to the head of .
Addition is commutative:
Addition is associative:
To subtract a vector, add its opposite:

(ii) Multiplication by scalar 

Multiplication of a vector by a positive scalar a, multiplies the magnitude but leaves the direction unchanged. (If a is negative, the direction is reversed.) Scalar multiplication is distributive:

(iii) Dot product of two vectors  

The dot product of two vectors is define by

where θ is the angle they form when placed tail to tail. Note that  is itself a scalar. The dot product is commutative,

and distributive, 

Geometrically  is the product of A times the projection of  along  (or the product  of B times the project ion of  along  ).

If the two vectors are parallel,

If two vectors are perpendicular, then

Law of cosines

Let  and then calculate dot product of  with itself.

C² = A² + B² - 2AB cosθ

(iv) Cross product of two vectors  

The cross product of two vectors is define by

where  is a unit vector(vector of length 1) point ing perpendicular to the plane of  and . Of course there are two directions perpendicular to any plane “in” and “out.” The ambiguity is resolved by the right-hand rule: let your fingers point in the direction of first vector and curl around (via the smaller angle) toward the second; then your thumb indicates the direction of . (In figure  points into the page;  points out of the page)

The cross product is distributive,

but not commutative. 
In fact,
Geometrically, || is the area of the parallelogram generated by  and  . If two vectors are parallel, their cross product is zero. 
In particular  for any vector .

Vector Algebra: Component Form

Let  and  be unit vectors parallel to the x, y and z axis, respectively. An arbitrary vector  can be expanded in terms of these basis vectors

The numbers A_x , A_y , and A_z are called component of  ; geometrically, they are the  projections of   along the three coordinate axes.

(i) Rule: To add vectors, add like components.

(ii) Rule: To multiply by a scalar, multiply each component.

Because  and  are mutually perpendicular unit vectors

Accordingly,

(iii) Rule: To calculate the dot product, multiply like components, and add.

In particular,
Similarly,

(iv) Rule: To calculate the cross product, form the determinant whose first row is ,, whose second row is  (in component form), and whose third row is .

Example 1: Find the angle between the face diagonals of a cube.

Solution: The face diagonals and are

So, ⇒
Also, ⇒

Example 2: Find the angle between the body diagonals of a cube.

Solution: The body diagonals and are

So, ⇒
Also,
⇒

Example 3: Find the components of the unit vector nˆ perpendicular to the plane shown in the figure.

Solution: The vectors   and  can be defined as

Triple Products

(i) Scalar triple product:

Geometrically  is the volume of the parallelepiped  generated by  and  , since  is the area of the base, and  is the altitude. Evidently,
In component form
Note that the dot and cross can be interchanged:

(ii) Vector triple product:
The vector triple product can be simplified by the so-called BAC-CAB rule: