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Class 11 Physics Vector Analysis NEET/JEE

VECTOR

On the basis of magnitude, direction and rules of addition, all physical quantities are classified into two groups as Scalars and Vectors:

Scalars

A scalar quantity is one whose specification is completed with its magnitude only. Two or more similar scalar quantities can be added according to the ordinary rules of the algebra.

For example mass, distance, speed, energy, electric flux, current electricity etc.

In the above example, current electricity has magnitude and direction but it is a scalar quantity because two different electrical currents can be added only with simple algebra, as

Vectors

A vector quantity is one whose specification is completed with its magnitude and direction. Two similar vector quantities can be added according to the law of parallelogram or triangle law. For example displacement, Velocity, acceleration, force, electric field intensity, current density etc.

A vector quantity can be represented by a line. The front end (arrowhead) represents the direction and length of the line gives its magnitude as:

OA = magnitude of the vector (not according to scalar). The magnitude of a vector can be written as, modulus of.

Types of Vectors

(a) Polar Vectors: A vector whose direction is along the direction of the motion of a body or particle is known as a polar vector.

  For example, Displacement, velocity, linear momentum, force etc. are Polar Vectors.

(b)   Axial Vectors: A vector whose direction is along the axis of rotation of the body or a particle is called an axial vector. An axial-vector always produces a rotational effect on the body.

For example, angular velocity, angular acceleration, torque, angular momentum are axial vectors.

(c) Equal vectors: Two vectors are said to be equal vectors if they have equal magnitude and same directions.

vectors and are equal vectors. i.e.,

(d) Co-initial vectors: Vectors having a common initial point are called co-initial vectors. The vectors A, B, C and D are said to be co-initial vectors.

(e)   Unit Vectors: A vector of unit magnitude and whose direction is the same as that of the given vector is called a unit vector. Basically unit vector represents the direction of the given considers a vector A. This vector can be written as :

Vector = (Magnitude of the vector) (direction of vector).

(f) The zero vector or Null vector: A vector whose magnitude is zero and no sense of direction is called zero vector or null vector. It is represented by

For Example: The position vector of origin, the acceleration of a particle moving with uniform velocity etc.

(i)  Addition or subtraction of zero vectors from a given vector does not affect the given vector.

For Example: The position vector of origin, the acceleration of a particle moving with uniform velocity etc.

(i) Addition or subtraction of zero vectors from a given vector does not affect the given vector.

Multiplication of a Vector by a Real Number The multiplication of a vector by a scalar quantity n gives a new vector whose magnitude is n times the magnitude of the given vector. Its direction is the same as that of the given vector if n is a positive real number. Suppose a vector

Additional Vectors

‚Äč(i) Addition of Collinear vector:

For Example: Suppose a body is displaced through 4 m due west and then it is further displaced through 6 m due west. Then the resultant displacement of the body = (4 m + 6 m). = 10 m due west.

(ii) Addition of two anti-collinear Vectors:

For Example: Suppose a body is displaced through 4 m due east and then 2 m due west

Resultant displacement vector

(iii) Addition of two vectors pointing in different directions: When two vectors are pointing in different directions, they can be added using laws of vector addition such as triangle law, parallelogram law and polygon law.

The triangle law of vector addition

If two vectors are represented both in magnitude and direction by the two sides of a triangle taken in the same order, then the resultant of these vectors is represented both in magnitude and direction by the third side of the triangle as shown below.

The resultant R can be calculated by the law of parallelogram.

Parallelogram Law of Vector Addition

The law of parallelogram for vector addition is applied when the two vectors act on the same point simultaneously at a certain angle.

Special Cases

Hence the resultant vector points in the direction of the given vectors.

i.e., the direction of the resultant vector is opposite to the direction of the vector whose magnitude is smaller.

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