Question

What is the dot product of I and j?

Answer 1

The dot product is also called scalar product because it gives a scalar quantity as a result of the dot product. It is also the product of the magnitude of the two given vectors and the cosine of the angle between them. It is the projection of one vector onto another vector, for parallel, the value of the dot product is one and for perpendicular, it is zero.
Formula used:
Suppose $\mathop A\limits^ \to$ and $\mathop B\limits^ \to$ are two vectors with components $\mathop A\limits^ \to = {A_x} + {A_y} + {A_z}$ and $\mathop B\limits^ \to = {B_x} + {B_y} + {B_z}$in space, then
$\mathop A\limits^ \to .\mathop B\limits^ \to = \left| A \right|\left| B \right|\cos \theta$
Where $\theta$ is the angle between the two vectors

Complete step-by-step solution:
The dot product of two vectors $\mathop A\limits^ \to$ and $\mathop B\limits^ \to$ with an angle $\theta$ is called the scalar product and it is equal to the product of the magnitude of the two vectors and cosine of the angle between them or it is the product of the magnitude of one of the vector and the component of the second vector.
So, $\mathop A\limits^ \to .\mathop B\limits^ \to = \left| A \right|\left| B \right|\cos \theta$
Or
$\mathop A\limits^ \to .\mathop B\limits^ \to = \left| A \right|(\left| B \right|\cos \theta )$
The quantity $A(Bcos\theta )$ is scalar.
Now in the given question, we are asked to find the dot product of $\widehat {i.}$ and $\widehat j$
Here $\widehat {i.}$ and $\widehat j$ are the unit vectors in the direction along the x-axis and y-axis respectively.
So, they are mutually perpendicular to each other, that's why $\theta = 90^\circ$.
Therefore,
$\widehat {i.}\widehat j = ij\cos {90^ \circ }$
$\widehat {i.}\widehat j = 0$.
Hence, the dot product of the two unit vectors $\widehat {i.}$ and $\widehat j$ is zero.

Note: The dot product is the product of the magnitude of the two given vectors and the cosine of the angle between them
similarly, $\widehat j.\widehat k = \widehat k.\widehat i = 0$ as they are mutually perpendicular to each other.
And $\widehat i.\widehat i = \widehat j.\widehat j = \widehat k.\widehat k = 1$.