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1)

If a,b,c are  distinct  real numbers are P,Q, R are three points  whose position  vectors are  respectively 

aˆi+bˆj+cˆk,bˆi+cˆj+aˆk and cˆi+aˆj+bˆk  , then QPR=


A) cos1(a+b+c)

B) π2

C) π3

D) cos1(a2+b2+c2abc)

Answer:

Option C

Explanation:

position vector of P= aˆi+bˆj+cˆk

Position vector of Q= bˆi+cˆj+aˆk

Position vector R=cˆi+aˆj+bˆk

Now, PQ= PV of Q-PV of P

 =(bˆi+cˆj+aˆk)-(aˆi+bˆj+cˆk)

 3172021484_d1.PNG

=(ba)ˆi+(cb)ˆj+(ac)ˆk

 Similarly, PR= (ca)ˆi+(ab)ˆj+(bc)ˆk

Now, PQ.PR=[(ba)ˆi+(cb)ˆj+(ac)ˆk].[(ca)ˆi+(ab)ˆj+(bc)ˆk]

    =(b-a)(c-a)+(c-b)(a-b)+(a-c)(b-c)

=a2+b2+c2abbcca

 |PQ|= (ba)2+(cb)2+(ac)2

|PR|= (ca)2+(ab)2+(bc)2

Now, cosθ=PQ.PR|PQ|.|PR|

 = a2+b2+c2abbcca2a2+2b2+2c22ab2bc2ca

 = a2+b2+c2abbcca2(a2+b2+c2abbcca)

cosθ=12

   θ= π3

 Here, QPR=π3