Question

How do you find the unit vector in the direction of the given vector of $v = i + j$?

Answer 1

Hint : By the definition of the unit vector they always have a magnitude which is also known as its modulus is one. It is also denoted by $\widehat a$ cap type sign on the desired vector. The unit vector can be found easily by dividing a vector by its magnitude the formula here can be written for $\overrightarrow a$ as
$\widehat {\overrightarrow v } = \dfrac{{\sqrt 2 }}{2}(\widehat i + \widehat j)$
Thus we will find the magnitude of our given actor and will then divide the calculated magnitude with the original vector. The magnitude can be calculated by using the formula by using the coefficients of its forming unit vectors $\widehat i$ and $\widehat j$ in this case here by squaring the coefficients and then finding the square root of their sum.
$\left| {\overrightarrow v } \right| = \sqrt {{1^2} + {1^2}}$ where $1$ and $1$ are the coefficient of the unit vectors $i$and $j$.

Complete step by step solution:
The unit vector is a vector whose magnitude is 1, we are given
$v = \widehat i + \widehat j$
We will first find the magnitude of the given vector , since $1$ and $1$ are the coefficients of the unit vectors in the given vector
$\left| {\overrightarrow v } \right| = \sqrt {{1^2} + {1^2}}$
$\left| {\overrightarrow v } \right| = \sqrt 2$
Now we will divide the given vector with its magnitude we just calculated so the unit vector of $\overrightarrow v$ is given by
$\widehat {\overrightarrow v } = \dfrac{{\widehat i + \widehat j}}{{\sqrt 2 }}$
Rationalizing the denominator we get
$\widehat {\overrightarrow v } = \dfrac{{\sqrt 2 }}{2}(\widehat i + \widehat j)$
Which is the required final answer

Note : Always remember that the unit vectors $\widehat i,\widehat j,\widehat k$ always point towards the $x,y,z$ axis in the three dimensional space. These three are called to be as the unit vector and are founding blocks in defining any vector; they are unit vectors so they have their magnitude to be always as $1$ .