Magnitude of vectors overrightarrow{ OA }, overrightarrow{ OB } and overrightarrow{ OC } in given fi

English

Gujarati

Hindi

3-1.Vectors

hard

The magnitude of vectors $\overrightarrow{ OA }, \overrightarrow{ OB }$ and $\overrightarrow{ OC }$ in the given figure are equal. The direction of $\overrightarrow{ OA }+\overrightarrow{ OB }-\overrightarrow{ OC }$ with $x$-axis will be

$\tan ^{-1} \frac{(1-\sqrt{3}-\sqrt{2})}{(1+\sqrt{3}+\sqrt{2})}$

$\tan ^{-1} \frac{(\sqrt{3}-1+\sqrt{2})}{(1+\sqrt{3}-\sqrt{2})}$

$\tan ^{-1} \frac{(\sqrt{3}-1+\sqrt{2})}{(1-\sqrt{3}+\sqrt{2})}$

$\tan ^{-1} \frac{(1+\sqrt{3}-\sqrt{2})}{(1-\sqrt{3}-\sqrt{2})}$

(JEE MAIN-2021)

Solution

Let magnitude be equal to $\lambda$.

$\overrightarrow{ OA }=\lambda\left[\cos 30^{\circ} \hat{ i }+\sin 30 \hat{ j }\right]=\lambda\left[\frac{\sqrt{3}}{2} \hat{ i }+\frac{1}{2} \hat{ j }\right]$

$\overrightarrow{ OB }=\lambda\left[\cos 60^{\circ} \hat{ i }-\sin 60 \hat{ j }\right]=\lambda\left[\frac{1}{2} \hat{ i }-\frac{\sqrt{3}}{2} \hat{ j }\right]$

$\overrightarrow{ OC }=\lambda\left[\cos 45^{\circ}(-\hat{ i })+\sin 45 \hat{ j }\right]=\lambda\left[-\frac{1}{\sqrt{2}} \hat{ i }+\frac{1}{\sqrt{2}} \hat{ j }\right]$

$\therefore \overrightarrow{ OA }+\overrightarrow{ OB }-\overrightarrow{ OC }$

$=\lambda\left[\left(\frac{\sqrt{3}+1}{2}+\frac{1}{\sqrt{2}}\right) \hat{ i }+\left(\frac{1}{2}-\frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}\right) \hat{ j }\right]$

$\therefore$ Angle with $x$-axis

$\tan ^{-1}\left[\frac{\frac{1}{2}-\frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}}{\frac{\sqrt{3}}{2}+\frac{1}{2}+\frac{1}{\sqrt{2}}}\right]=\tan ^{-1}\left[\frac{\sqrt{2}-\sqrt{6}-2}{\sqrt{6}+\sqrt{2}+2}\right]$

$=\tan ^{-1}\left[\frac{1-\sqrt{3}-\sqrt{2}}{\sqrt{3}+1+\sqrt{2}}\right]$

Standard 11

Physics

At what angle should the two forces $2 P$ and $\sqrt{2} P$ act so that the resultant force is $P \sqrt{10}$ ?

medium

$\overrightarrow A \, = \,3\widehat i\, + \,2\widehat j$ , $\overrightarrow B \, = \widehat {\,i} + \widehat j – 2\widehat k$ then find their addition by algebric method.

easy

Two forces $P$ and $Q$, of magnitude $2F$ and $3F$, respectively, are at an angle $\theta $ with each other. If the force $Q$ is doubled, then their resultant also gets doubled. Then, the angle $\theta $ is ……. $^o$

medium

(JEE MAIN-2019)

The angle between vector $\vec{Q}$ and the resultant of $(2 \overrightarrow{\mathrm{Q}}+2 \overrightarrow{\mathrm{P}})$ and $(2 \overrightarrow{\mathrm{Q}}-2 \overrightarrow{\mathrm{P}})$ is:

medium

(JEE MAIN-2024)

Given $a+b+c+d=0,$ which of the following statements eare correct:

$(a)\;a, b,$ c, and $d$ must each be a null vector,

$(b)$ The magnitude of $(a+c)$ equals the magnitude of $(b+d)$

$(c)$ The magnitude of a can never be greater than the sum of the magnitudes of $b , c ,$ and $d$

$(d)$ $b + c$ must lie in the plane of $a$ and $d$ if $a$ and $d$ are not collinear, and in the line of a and $d ,$ if they are collinear ?

medium

The magnitude of vectors $\overrightarrow{ OA }, \overrightarrow{ OB }$ and $\overrightarrow{ OC }$ in the given figure are equal. The direction of $\overrightarrow{ OA }+\overrightarrow{ OB }-\overrightarrow{ OC }$ with $x$-axis will be

Solution

Similar Questions

At what angle should the two forces $2 P$ and $\sqrt{2} P$ act so that the resultant force is $P \sqrt{10}$ ?

$\overrightarrow A \, = \,3\widehat i\, + \,2\widehat j$ , $\overrightarrow B \, = \widehat {\,i} + \widehat j – 2\widehat k$ then find their addition by algebric method.

Two forces $P$ and $Q$, of magnitude $2F$ and $3F$, respectively, are at an angle $\theta $ with each other. If the force $Q$ is doubled, then their resultant also gets doubled. Then, the angle $\theta $ is ……. $^o$

The angle between vector $\vec{Q}$ and the resultant of $(2 \overrightarrow{\mathrm{Q}}+2 \overrightarrow{\mathrm{P}})$ and $(2 \overrightarrow{\mathrm{Q}}-2 \overrightarrow{\mathrm{P}})$ is:

Start a Free Trial Now