Question

What does the \[\hat{i}\] and $\hat{j}$ stand for in vectors?

Answer 1

Hint: We know that a vector is an object that has both a magnitude and a direction. The direction of the vectors is from its tail to its head where the head of a vector is the arrow mark and the tail is the line behind it.

Complete step-by-step solution:
Let us consider the vectors. Suppose we are dealing with the vectors that lie in the $xy-$plane.
In this case, we will have two orthogonal unit vectors that are lying in each $x$ and $y$ directions.
So, the unit vector that lies along the $x$ direction is $\hat{i}$ and the unit vector that lies along the $y$ direction is $\hat{j}.$
As we already know, the $x$ axis and the $y$ axis are perpendicular to each other. So, the unit vectors along these axes are also perpendicular to each other.
We call the vectors that are perpendicular to each other the orthogonal vectors.
As we know, each vector has a magnitude as well as a direction.
Since we discussed the directions, we are left with the magnitudes.
As we have defined, \[\hat{i}\] and $\hat{j}$ are unit vectors. This simply implies that the magnitude of the unit vectors is equal to $1.$
We can say that the magnitude of \[\hat{i}\] and $\hat{j}$ are equal to $1.$
Hence \[\hat{i}\] and $\hat{j}$ are orthogonal unit vectors along $x$ and $y$ axes respectively.

Note: A scalar is an object that does not have a direction but that has a magnitude. A set of vectors are orthonormal vectors if all vectors in the set are mutually perpendicular to each other.