Question

13. Given that P= Q = R. If P+ = Ř then the
angle between and © is 0,. If
P+ + = 7 then the angle between 7 and
Ris 02. Then the relation between 0, and en
is​

Answer 1

☆Question:

Given that P= Q = R. If $\vec{P}+\vec{Q}= \vec{R}$ then the angle between $\vec{P}\;and\; \vec{Q}$ is $\theta_1$. If $\vec{P}+\vec{Q}+\vec{R}=0$ then the angle between $\vec{P}\;and\; \vec{R}$ is $\theta_2$. Then the relation between $\theta_1$, and $\theta_2$ is

☆Solution:

kindly see the attached figure!

Case 1-

$\vec{P}+\vec{Q}= \vec{R}$

simply do the vector addition.

$\rightarrow\rm \sqrt{P^2+Q^2 + 2PQcos\theta_1} = R$

$\rightarrow\rm P^2+Q^2 + 2PQcos\theta_1 = R^2$

☆Its given that P=Q=R

$\rightarrow\rm P^2+P^2 + 2P^2cos\theta_1 = P^2$

$\rightarrow\rm 2P^2 + 2P^2cos\theta_1 = P^2$

$\rightarrow\rm P^2= -2P^2cos\theta_1$

$\rightarrow\rm cos\theta_1 = \dfrac{-1}{2}$

$\rightarrow\bf \theta_1 = 120\degree$

Case 2-

$\vec{P}+\vec{Q}+\vec{R}=0$

$\implies \vec{P}+\vec{R}= \vec{-Q}$

Again do the vector addition between P and R.

$\rightarrow\rm \sqrt{P^2+R^2 + 2PQcos\theta_2} = -R$

$\rightarrow\rm P^2+Q^2 + 2PQcos\theta_2 = R^2$

$\rightarrow\rm P^2+P^2 + 2P^2cos\theta_2 = P^2$

$\rightarrow\rm 2P^2 + 2P^2cos\theta_2 = P^2$

$\rightarrow\rm P^2= -2P^2cos\theta_2$

$\rightarrow\rm cos\theta_2 = \dfrac{-1}{2}$

$\rightarrow\bf \theta_2 = 120\degree$

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☆Relation

$\bf \theta_1 = \theta_2 =120\degree$

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