Answer

Verified

338.4k+ views

**Hint:**First, simplified and shift the vectors by connecting the tails of each vector and then break the vectors in the rectangular components in the form of $\hat i$ and $\hat j$ finally obtain the vectors sum of the vectors and the magnitude of the sum.

**Complete step by step answer:**

From the question, we know that the magnitude of the three vectors are $\vec A = 1,\;\vec B = 2$ and $\vec C = 3$ and these three vectors forms an equilateral triangle.We know that the all the angles of the equilateral triangle is equal to $60^\circ $ and the figure below represents the three vectors and the simplified representation of these three vectors.

Now write the all vectors in terms of direction $\hat i$ and $\hat j$.

From the figure, we know that the vectors $\vec B$ and $\vec C$ are at angle $60^\circ $ so we will split them into the rectangular components.The vector $\vec A$ is written as,

$\vec A = 1\hat i$

The vector $\vec B$ is written as,

$\vec B = - \vec B\cos 60^\circ \hat i + \vec B\sin 60^\circ \hat j$

Substitute the values in the above equation, we get,

$\vec B = - \left( 2 \right)\dfrac{1}{2}\hat i + \left( 2 \right)\dfrac{{\sqrt 3 }}{2}\hat j\\

\Rightarrow\vec B = - 1\hat i + \sqrt 3 \hat j

$

The vector $\vec C$ is written as,

$\vec C = - \vec C\cos 60^\circ \hat i - \vec C\sin 60^\circ \hat j$

Substitute the values in the above equation, we get,

$

\vec C = - \left( 3 \right)\dfrac{1}{2}\hat i - \left( 3 \right)\dfrac{{\sqrt 3 }}{2}\hat j\\

\Rightarrow\vec C = - \dfrac{3}{2}\hat i - \dfrac{{3\sqrt 3 }}{2}\hat j

$

Now we calculate the sum of the three vectors, we have,

$

\vec A + \vec B + \vec C = \left( {1\hat i} \right) + \left( { - 1\hat i + \sqrt 3 \hat j} \right) + \left( { - \dfrac{3}{2}\hat i - \dfrac{{3\sqrt 3 }}{2}\hat j} \right)\\

\Rightarrow\vec A + \vec B + \vec C = - \dfrac{3}{2}\hat i - \dfrac{{3\sqrt 3 }}{2}\hat j

$

Now, we calculate the magnitude of the three vectors,

$\left| {\vec A + \vec B + \vec C} \right| = \left| { - \dfrac{3}{2}\hat i - \dfrac{{3\sqrt 3 }}{2}\hat j} \right|$

After simplifying the above equation, we have,

\[

\left| {\vec A + \vec B + \vec C} \right| = \sqrt {{{\left( { - \dfrac{3}{2}} \right)}^2} + {{\left( { - \dfrac{{3\sqrt 3 }}{2}} \right)}^2}} \\

\Rightarrow\left| {\vec A + \vec B + \vec C} \right|= \sqrt {\dfrac{9}{4} + \dfrac{3}{4}} \\

\therefore\left| {\vec A + \vec B + \vec C} \right|= \sqrt {\dfrac{{12}}{4}}

\]

After simplifying the above equation, we get $\sqrt 3 $.

**Thus, the magnitude of the resultant of three vectors of magnitude 1, 2 and 3 is $\sqrt 3 $ and option D is correct.**

**Note:**Make sure the triangles are formed by connecting the head of one vector with the tail of another vector and while splitting the vectors in the rectangular components, make sure you add minus in for leftwards and downward direction. Only $\hat i$ quantity will add with $\hat i$ quantity not $\hat j$ quantity.

Recently Updated Pages

Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred

What is the stopping potential when the metal with class 12 physics JEE_Main

The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main

Basicity of sulphurous acid and sulphuric acid are

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Which places in India experience sunrise first and class 9 social science CBSE

The list which includes subjects of national importance class 10 social science CBSE

What is pollution? How many types of pollution? Define it

State the laws of reflection of light