Let A, B and C be three points on a circle of | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the vectors be $\vec{OA}, \vec{OB}, \vec{OC}$. We need to find the magnitude of $\vec{R} = \vec{OA} + \vec{OB} + \vec{OC}$.Let $\vec{OB}$ be a

Vedclass · Accepted Answer

$(\sqrt{2}+1) R$

Vedclass · Answer

(B) Let the vectors be $\vec{OA}, \vec{OB}, \vec{OC}$. We need to find the magnitude of $\vec{R} = \vec{OA} + \vec{OB} + \vec{OC}$.Let $\vec{OB}$ be along the $y$-axis,so $\vec{OB} = R\hat{j}$.Then $\vec{OA} = R(\cos 45^{\circ} \hat{i} + \sin 45^{\circ} \hat{j}) = R(\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j})$.And $\vec{OC} = R(\cos 45^{\circ} \hat{i} - \sin 45^{\circ} \hat{j}) = R(\frac{1}{\sqrt{2}}\hat{i} - \frac{1}{\sqrt{2}}\hat{j})$.Summing these vectors:$\vec{R} = \vec{OA} + \vec{OB} + \vec{OC} = R(\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j}) + R\hat{j} + R(\frac{1}{\sqrt{2}}\hat{i} - \frac{1}{\sqrt{2}}\hat{j})$$\vec{R} = R(\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}})\hat{i} + R(\frac{1}{\sqrt{2}} + 1 - \frac{1}{\sqrt{2}})\hat{j}$$\vec{R} = R(\frac{2}{\sqrt{2}})\hat{i} + R\hat{j} = \sqrt{2}R\hat{i} + R\hat{j}$.The magnitude is $|\vec{R}| = \sqrt{(\sqrt{2}R)^2 + R^2} = \sqrt{2R^2 + R^2} = \sqrt{3R^2} = \sqrt{3}R$.Wait,re-evaluating the geometry: The resultant of $\vec{OA}$ and $\vec{OC}$ is $2R \cos(45^{\circ}) \hat{j} = \sqrt{2}R \hat{j}$.Adding $\vec{OB} = R \hat{j}$,the total resultant is $(\sqrt{2} + 1)R \hat{j}$.Thus,the magnitude is $(\sqrt{2} + 1)R$.

Let $A, B$ and $C$ be three points on a circle of radius $R$. If $O$ is the centre of the circle and $\angle AOB = 45^{\circ}, \angle BOC = 45^{\circ}$,then the resultant of $\vec{OA}, \vec{OB}$ and $\vec{OC}$ has magnitude

Similar Questions

Find the position vector of the point which divides the join of the points with position vectors $2\vec{a} - 3\vec{b}$ and $3\vec{a} - 2\vec{b}$ externally in the ratio $2:3$.

Let $P, Q, R,$ and $S$ be points on a plane with position vectors $-2\hat{i} - \hat{j}$,$4\hat{i}$,$3\hat{i} + 3\hat{j}$,and $-3\hat{i} + 2\hat{j}$ respectively. What type of quadrilateral is $PQRS$?

Let $\vec{a}=3 \hat{i}+\hat{j}-2 \hat{k}$,$\vec{b}=-5 \hat{i}+7 \hat{j}$,and $\vec{c}=3 \hat{i}+y \hat{j}$ be three vectors such that $|\vec{a}-\vec{b}+\vec{c}|=\sqrt{141}$. If $y_1$ and $y_2$ are the values of $y$ satisfying the given condition,then $|y_1-y_2|=$

Given vectors $\vec{a} = 2 \hat{i} - \hat{j} + 2 \hat{k}$ and $\vec{b} = -\hat{i} + \hat{j} - \hat{k}$. The vector in the direction of $\vec{a} + \vec{b}$ with magnitude $\sqrt{2}$ is . . . . . . .

The angle made by the position vector of the point $(5, -4, -3)$ with the positive direction of the $X$-axis is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module