General Vectors: Vector Equality, Unit Vector

A vector, as we know, is any object that has both a direction and magnitude. Here we will learn about when two vectors are considered equal, what happens when a vector is multiplied with a number and what is called a unit vector.

Equality of Vectors:

If we want to talk about equality between two vectors, they have to represent the same physical quantity. If their magnitudes, directions and other characteristics are identical, two vectors will be considered equal. We can therefore conclude that a parallel vector translation does not cause any changes in the vector.

Multiplication of a vector by a Number:

Let’s say c is considered as a vector of magnitude ‘c’ and a number is denoted by n.  If we take the product of  c  and n, then we will receive a new vector. Let’s call this new vector as   d.

The definition of vector d= n c will be a vector whose magnitude is nc. Vector  d will be of the same direction as vector c which means that if the vector c is in the negative direction of ‘x’ then vector d will also be in the same direction and if the vector c is in positive direction of ‘x’ then, vector  d will be in the positive direction of ‘x’.

If n is negative, then the direction of  c will be opposite or contradictory to the direction of vector d . Should n equal -1, then the product with n will simply reverse the movement direction of the vector.

Unit Vector:

A vector of length or magnitude of 1 unit is considered a unit vector. Hence, they are used to depict the vector quantity’s direction within various systems of coordinates. Let’s take Cartesian coordinates for example:

j = unit vector in y direction

i = unit vector in x direction

k  = unit vector in z direction

Using Cartesian coordinates the position vectors can be denoted by the following:

r= xi+yj+xk