# If a,b and c are unit vectors then $\left| a-b{{|}^{2}} \right.+|b-c{{|}^{2}}+|c-a{{|}^{2}}$ does not exceed

A. 4

B. 9

C. 8

D. 6

Answer

Verified

91.2k+ views

**Hint:**We are given a, b, and c as unit vectors, and we must find the value that the given equation does not exceed. A unit vector is a vector with a magnitude of one. To answer this question, we use the whole square formula, and we get our desired answer by entering the values into the formulas and solving the equations.

**Formula Used**:

${{(a-b)}^{2}}+{{(b-c)}^{2}}+{{(c-a)}^{2}}=2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})-2(a.b+b.c+c.a)$

**Complete step by step solution:**

We are given that a , b and c are the unit vectors , then

$\left| a-b{{|}^{2}} \right.+|b-c{{|}^{2}}+|c-a{{|}^{2}}$-------------------------(1)

We know the formula of ${{(a-b)}^{2}}+{{(b-c)}^{2}}+{{(c-a)}^{2}}=2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})-2(a.b+b.c+c.a)$

Put the above values in equation (1) and we get

$\left| a-b{{|}^{2}} \right.+|b-c{{|}^{2}}+|c-a{{|}^{2}}$ = $2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})-2(a.b+b.c+c.a)$

By putting the values in above equation, we get

$2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})-2(a.b+b.c+c.a)$ = 2$\times 3 – 2 (a.b + b.c + c.a )$ ----------- (2)

By simplifying $2 (a.b + b.c + c.a ) = \left\{ {{(a+b+c)}^{2}}-{{a}^{2}}-{{b}^{2}}-{{c}^{2}} \right\}$

By putting the values in equation (2), we get

$2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})-2(a.b+b.c+c.a)$ = 6 – $\left\{ {{(a+b+c)}^{2}}-{{a}^{2}}-{{b}^{2}}-{{c}^{2}} \right\}$

By solving the above equation, we get 9 - $|a+b+c{{|}^{2}}\le 9$

Hence, value of $\left| a-b{{|}^{2}} \right.+|b-c{{|}^{2}}+|c-a{{|}^{2}}$ does not exceed 9

**Option ‘B’ is correct**

**Note:**Vector units have both direction and magnitude. However, sometimes one is interested only in direction and not the magnitude. Vectors are often considered unit length. These unit vectors are generally used to represent direction, with a scalar coefficient providing the magnitude.

Last updated date: 04th Jun 2023

•

Total views: 91.2k

•

Views today: 1.47k

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

What is the Full Form of PVC, PET, HDPE, LDPE, PP and PS ?

Alfred Wallace worked in A Galapagos Island B Australian class 12 biology CBSE

Imagine an atom made up of a proton and a hypothetical class 12 chemistry CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

How do you define least count for Vernier Calipers class 12 physics CBSE

Why is the cell called the structural and functional class 12 biology CBSE

A 30 solution of H2O2 is marketed as 100 volume hydrogen class 11 chemistry JEE_Main