Question

The unit vector along $\hat{i}+\hat{j}$ is
(A). $\hat{k}$
(B). $\hat{i}+\hat{j}$
(C). $\dfrac{\hat{i}+\hat{j}}{\sqrt{2}}$
(D). $\dfrac{\hat{i}+\hat{j}}{2}$

Answer 1

Hint: Vectors are quantities which possess magnitude and direction. Vectors are also used to represent position or direction in space. The given vector represents the position of a point in space. The magnitude of a point is defined as the distance of a point from the origin in space. A vector is converted into a unit vector by dividing the vector by its magnitude.

Formula used:
$\hat{a}=\dfrac{{{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k}}{\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}}$

Complete answer:
Vector quantities are those quantities that have magnitude as well as direction. Vectors are used to represent the direction in space and its length is used to represent the magnitude. The unit vectors $\hat{i}$, $\hat{j}$ and $\hat{k}$ are used to represent the vector in space in terms of three fundamental axes; x-axis, y-axis and z-axis.
The vector whose magnitude is one is known as a unit vector
Let us assume a vector, $\vec{a}={{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k}$
The magnitude of the vector is calculated as
$\left| {\vec{a}} \right|=\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$
To convert a vector into a unit vector, we divide it by its magnitude. Therefore, unit vector of $\vec{a}={{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k}$ will be-
$\hat{a}=\dfrac{{{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k}}{\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}}$
Therefore, the unit vector is $\dfrac{{{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k}}{\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}}$.
Given, vector $\vec{t}=\hat{i}+\hat{j}$, similarly the unit vector of $\vec{t}$ will be-
$\begin{align}
  & \hat{t}=\dfrac{\hat{i}+\hat{j}}{\sqrt{{{1}^{2}}+{{1}^{2}}}} \\
 & \therefore \hat{t}=\dfrac{\hat{i}+\hat{j}}{\sqrt{2}} \\
\end{align}$
The unit vector of $\vec{t}$ is $\dfrac{\hat{i}+\hat{j}}{\sqrt{2}}$.
Therefore, the unit vector along $\hat{i}+\hat{j}$ is $\dfrac{\hat{i}+\hat{j}}{\sqrt{2}}$.

Hence, the correct option is (C).

Note:
The other type of quantities is scalar quantities that only have magnitude. Their magnitude is represented by the unit. Different mathematical operations can be applied on vectors like addition, subtraction, multiplication etc. The vectors which tell us the position of a point in space are known as position vectors and they can be interconverted between the coordinate system and vectors.