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  • Explain solution RD Sharma class 12 Chapter 22 Algebra of Vectors Exercise 22.6 question 8

Explain solution RD Sharma class 12 Chapter 22 Algebra of Vectors Exercise 22.6 question 8

Answers (1)

Answer:

\begin{aligned} &|A \vec{B}|=\sqrt{\left(b_{1}-a_{1}\right)^{2}+\left(b_{2}-a_{2}\right)^{2}+\left(b_{3}-a_{3}\right)^{2}} \\\\ &|B \vec{C}|=\sqrt{\left(c_{1}-b_{1}\right)^{2}+\left(c_{2}-b_{2}\right)^{2}+\left(c_{3}-b_{3}\right)^{2}} \\\\ &|C \vec{A}|=\sqrt{\left(a_{1}-c_{1}\right)^{2}+\left(a_{2}-c_{2}\right)^{2}+\left(a_{3}-c_{3}\right)^{2}} \end{aligned}

Hint:

Use position vector formula.

Given:

a_{1} \hat{\imath}+a_{2} \hat{\jmath}+a_{3} \hat{k}, b_{1} \hat{\imath}+b_{2} \hat{\jmath}+b_{3} \hat{k} \text { and } c_{1} \hat{\imath}+c_{2} \hat{\jmath}+c_{3} \hat{k}

Solution:

Let the position vector of the vertex A is  a_{1} \hat{\imath}+a_{2} \hat{j}+a_{3} \hat{k}

And similarly  B=b_{1} \hat{\imath}+b_{2} \hat{\jmath}+b_{3} \hat{k} \text { and } C=c_{1} \hat{\imath}+c_{2} \hat{\jmath}+c_{3} \hat{k}

Side AB is

\begin{aligned} &A \vec{B}=\vec{B}-\vec{A}=\left(b_{1} \hat{\imath}+b_{2} \hat{\jmath}+b_{3} \hat{k}\right)-\left(a_{1} \hat{\imath}+a_{2} \hat{\jmath}+a_{3} \hat{k}\right)\\\\ &A \vec{B}=\left(b_{1}-a_{1}\right) \hat{\imath}+\left(b_{2}-a_{2}\right) \hat{\jmath}+\left(b_{3}-a_{3}\right) \hat{k} \end{aligned}  Eq.(i)
 

Equation ( i ) vector representation of the side AB

Magnitude of side AB,

|A \vec{B}|=\sqrt{\left(b_{1}-a_{1}\right)^{2}+\left(b_{2}-a_{2}\right)^{2}+\left(b_{3}-a_{3}\right)^{2}}

And similarly for side BC and CA

\begin{aligned} &B \vec{C}=\vec{C}-\vec{B}=\left(c_{1} \hat{\imath}+c_{2} \hat{\jmath}+c_{3} \hat{k}\right)-\left(b_{1} \hat{\imath}+b_{2} \hat{\jmath}+b_{3} \hat{k}\right)\\\\ &B \vec{C}=\left(c_{1}-b_{1}\right) \hat{\imath}+\left(c_{2}-b_{2}\right) \hat{\jmath}+\left(c_{3}-b_{3}\right) \hat{k} \end{aligned}       Eq.(ii)
 

\begin{aligned} &C \vec{A}=\vec{A}-\vec{C}=\left(a_{1} \hat{\imath}+a_{2} \hat{\jmath}+a_{3} \hat{k}\right)-\left(c_{1} \hat{\imath}+c_{2} \hat{\jmath}+c_{3} \hat{k}\right) \\\\ &C \vec{A}=\left(a_{1}-c_{1}\right) \hat{\imath}+\left(a_{2}-c_{2}\right) \hat{\jmath}+\left(a_{3}-c_{3}\right) \hat{k} \end{aligned}

Length of side BC and CA

\begin{aligned} &|B \vec{C}|=\sqrt{\left(c_{1}-b_{1}\right)^{2}+\left(c_{2}-b_{2}\right)^{2}+\left(c_{3}-b_{3}\right)^{2}} \\\\ &|C \vec{A}|=\sqrt{\left(a_{1}-c_{1}\right)^{2}+\left(a_{2}-c_{2}\right)^{2}+\left(a_{3}-c_{3}\right)^{2}} \end{aligned}

 

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