Question

The unit vector along $\mathop i\limits^ \wedge + \mathop j\limits^ \wedge$ is:
A. $\mathop k\limits^ \wedge$
B. $\mathop i\limits^ \wedge + \mathop j\limits^ \wedge$
C. $\dfrac{{\mathop i\limits^ \wedge + \mathop j\limits^ \wedge }}{{\sqrt 2 }}$
D. $\dfrac{{\mathop i\limits^ \wedge + \mathop j\limits^ \wedge }}{2}$

Answer 1

Vectors have both the magnitude and direction. We have got the formula to find out the magnitude of a vector. After finding the magnitude of a vector if we divide the given vector expression with its magnitude then we will get the unit vector for the given vector. Unit vector tells us the direction of the given vector.

Formula used:
$\mathop r\limits^ \to = \left| r \right|\mathop r\limits^ \wedge$

Complete step-by-step answer:
In order to represent a vector first we should have one coordinate system. There will be one reference point called origin. There will be 2 reference axes if we consider only a two dimensional plane and there will be three reference axes if we consider three dimensional space.
All these axes will be mutually perpendicular. The point of intersection of these axes will be the origin.
Let $\mathop r\limits^ \to$ be the vector and this vector will have both magnitude and direction. Magnitude of vector will be $\left| r \right|$ while the direction of this vector will be along $\mathop r\limits^ \wedge$. $\mathop r\limits^ \wedge$ is called as unit vector
From the formula $\mathop r\limits^ \to = \left| r \right|\mathop r\limits^ \wedge$ we get unit vector as $\dfrac{{\mathop r\limits^ \to }}{{\left| r \right|}} = \mathop r\limits^ \wedge$.
If $\mathop r\limits^ \to = a\mathop i\limits^ \wedge + b\mathop j\limits^ \wedge$ then magnitude of vector $\left| r \right|$ will be given as $\left| r \right| = \sqrt {{a^2} + {b^2}}$
Hence unit vector will be
\eqalign{ & \mathop r\limits^ \wedge = \dfrac{{\mathop r\limits^ \to }}{{\left| r \right|}} \cr & \Rightarrow \mathop r\limits^ \wedge = \dfrac{{a\mathop i\limits^ \wedge + b\mathop j\limits^ \wedge }}{{\sqrt {{a^2} + {b^2}} }} \cr}
For vector $\mathop i\limits^ \wedge + \mathop j\limits^ \wedge$ unit vector will be
\eqalign{ & \mathop r\limits^ \wedge = \dfrac{{a\mathop i\limits^ \wedge + b\mathop j\limits^ \wedge }}{{\sqrt {{a^2} + {b^2}} }} \cr & \Rightarrow \mathop r\limits^ \wedge = \dfrac{{\mathop i\limits^ \wedge + \mathop j\limits^ \wedge }}{{\sqrt {{1^2} + {1^2}} }} \cr & \Rightarrow \mathop r\limits^ \wedge = \dfrac{{\mathop i\limits^ \wedge + \mathop j\limits^ \wedge }}{{\sqrt 2 }} \cr}
Hence answer will be option C

So, the correct answer is “Option C”.

Additional Information: Every physical quantity which has direction need not be a vector. There are some quantities which we consider have direction but can’t include them in vectors. They are pressure, current and velocity of light. They don’t follow the vector law of addition hence they can’t be called vectors.

Note: For a particular vector, direction is given by the unit vector. Even though if we multiply the unit vector with some magnitude or number the length of the vector will increase but the direction will remain the same.