Science Articles

WHAT ARE THE TYPES AND PROPERTIES OF VECTORS?

In our daily life, we generally come across two types of quantities, namely scalars and vectors.
Scalars: a quantity that has magnitude only is known as a scalar.
Examples: Each of the quantities mass, length, time etc are scalars.


Warning: count(): Parameter must be an array or an object that implements Countable in /home/steml1og/public_html/justscience.in/wp-includes/media.php on line 1206

In our daily life, we generally come across two types of quantities, namely scalars and vectors.

Scalars: a quantity that has magnitude only is known as a scalar.

Examples:  Each of the quantities mass, length, time etc are scalars.

Vectors: A quantity that has magnitude as well as direction is known as a vector.

Examples: each of these quantities force, velocity, acceleration is a vector.

We define vector as “a directed line segment”

A directed line segment with an initial point A and a terminal point B is a vector donated by  .

The magnitude of  is denoted by | |.

Some commonly used terminology is given below

Unit vector: A vector    .

Equal vectors: Two vectors  and  are said to be equal if they have the same magnitude and the same direction regardless of the positions of their initial points.

Negative of a vector : A vector having the same magnitude as that of a given vector and a direction opposite to that of the is known as the negative of the given vector , to be denoted by .

Thus if . = , then  = –

Zero or null vector: A vector whose initial and terminal points coincide is known as a zero vector, denoted by  .

Cleary the magnitude of a zero vector is zero but it cannot be assigned any definite direction.

Thus,  =

Coinitial vectors: two or more vectors having the same initial points are known as coinitial vectors.

Collinear vectors: two vectors having the same or parallel supports are known as collinear vectors.

Like Vector: Collinear vectors having the same direction are known as like vectors.

Unlike Vectors: Collinear vectors having opposite directions are known as unlike vectors.

Free Vectors: If the initial point of a vector is not specified it is known as a free vector.

Localized Vector: I vector drawn parallel to a given vector through a specified point as the initial point is known as localized vector.

Coplanar Vectors: three or more non zero vectors lying in the same plane or parallel to the same plane are said to be coplanar, otherwise they are known as non-coplanar.

Position of a vector: let O be the origin of and let A be a point such that  = , then we say that the position vector of A is .

Algebraic Properties of Vectors

  • Commutative (vector) P + Q = Q + P
  • Associative (vector) (P + Q) + R = P + (Q + R)
  • Additive identity There is a vector 0 such
    that (P + 0) = P = (0 + P)
    for all P
  • Additive inverse For any P there is a vector -P such that P + (-P) = 0
  • Distributive (vector) r(P + Q) = rP + rQ
  • Multiplicative identity: For the real number 1, 1.P = P for each P

Scalar product of two vectors

Let a and b be two vectors and let ? be the angle between them. Then, the scalar product or dot product of a and b is defined as

a.b =|a| |b| cos ? = abcos?

Vector product of two vectors

Let a and b be two nonzero, nonparallel vectors, and let ? be the angle between them such that 0< ?<?.

Then, the vector product of a and b is defined as

a * b= (|a| |b| sin ?)