Question

Find the unit vector in the direction of vector $\overrightarrow {PQ}$ , where P and Q are the points $\left( {1,2,3} \right)$ and $\left( {4,5,6} \right)$ respectively.

Answer 1

Hint : Unit vector is the vector with magnitude 1. Unit vector between the point A and B is given by the formula $\overrightarrow {AB} = \dfrac{{\overrightarrow {AB} }}{{\left| {AB} \right|}}$
Where $\left| {AB} \right|$ is the magnitude of the vector (A,B) given by the formula $\left| {AB} \right| = \sqrt {{A^2} + {B^2}}$ and $\overrightarrow {AB}$ is the vector between the point A and B.
In this question we are given with two points whose unit vector is to be found, so first we will find the vector between the point P and Q and also their magnitude and then by substituting those values in the unit vector formula we will find the unit vector in the direction of vector PQ.

Complete step-by-step answer :
Given the point between which unit vector is to be found is

$$P \to \left( {1,2,3} \right) \\\ Q \to \left( {4,5,6} \right) \;$$

Now we will find the vector joining the points P and Q, we can write

$$\overrightarrow {PQ} = \left( {4 - 1} \right)\hat i + \left( {5 - 2} \right)\hat j + \left( {6 - 3} \right)\hat k \\\ = 3\hat i + 3\hat j + 3\hat k \;$$

Therefor the vector joining P and Q is
$\overrightarrow {PQ} = 3\hat i + 3\hat j + 3\hat k$
Now let us find the magnitude of the vector joining P and Q to find the unit vector, hence we can write this as
$Magnitude\left( {\overrightarrow {PQ} } \right) = 3\hat i + 3\hat j + 3\hat k$
So the magnitude of the vector will be

$$\left| {\overrightarrow {PQ} } \right| = \sqrt {{3^2} + {3^2} + {3^2}} \\\ = \sqrt {9 + 9 + 9} \\\ = \sqrt {27} \\\ = 3\sqrt 3 \;$$

Hence the unit vector in the direction of
$\overrightarrow {PQ} = \dfrac{{\overrightarrow {PQ} }}{{\left| {PQ} \right|}} = \dfrac{{3\hat i + 3\hat j + 3\hat k}}{{3\sqrt 3 }} = \dfrac{{\hat i}}{{\sqrt 3 }} + \dfrac{{\hat j}}{{\sqrt 3 }} + \dfrac{{\hat k}}{{\sqrt 3 }}$
So, the correct answer is “$\dfrac{{\hat i}}{{\sqrt 3 }} + \dfrac{{\hat j}}{{\sqrt 3 }} + \dfrac{{\hat k}}{{\sqrt 3 }}$ ”.

Note : A unit vector has the magnitude of 1. If two vectors of equal magnitude pointing in opposite directions will have sum as zero and if the magnitude of two vectors are not equal then their sum can never be zero.
$\hat i$ represent unit vector in positive x-axis direction
$\hat j$ represent unit vector in positive y-axis direction