Two vectors overrightarrow{A} and overrightar | vedclass

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

Question

(C) Let the magnitudes of vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ be $A = B = a$.The magnitude of the sum is given by $|\overrightarrow{

Vedclass · Accepted Answer

$\cos^{-1}\left(\frac{3}{5}\right)$

Vedclass · Answer

(C) Let the magnitudes of vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ be $A = B = a$.The magnitude of the sum is given by $|\overrightarrow{A} + \overrightarrow{B}| = \sqrt{a^2 + a^2 + 2a^2 \cos 	heta} = \sqrt{2a^2(1 + \cos 	heta)} = \sqrt{4a^2 \cos^2(	heta/2)} = 2a \cos(	heta/2)$.The magnitude of the difference is given by $|\overrightarrow{A} - \overrightarrow{B}| = \sqrt{a^2 + a^2 - 2a^2 \cos 	heta} = \sqrt{2a^2(1 - \cos 	heta)} = \sqrt{4a^2 \sin^2(	heta/2)} = 2a \sin(	heta/2)$.According to the problem,$|\overrightarrow{A} + \overrightarrow{B}| = 2|\overrightarrow{A} - \overrightarrow{B}|$.Substituting the expressions,we get $2a \cos(	heta/2) = 2(2a \sin(	heta/2))$.This simplifies to $\cos(	heta/2) = 2 \sin(	heta/2)$,or $	an(	heta/2) = 1/2$.Using the identity $\cos 	heta = \frac{1 - 	an^2(	heta/2)}{1 + 	an^2(	heta/2)}$,we substitute $	an(	heta/2) = 1/2$:$\cos 	heta = \frac{1 - (1/2)^2}{1 + (1/2)^2} = \frac{1 - 1/4}{1 + 1/4} = \frac{3/4}{5/4} = \frac{3}{5}$.Therefore,$	heta = \cos^{-1}(3/5)$.

Similar Questions

Maximum and minimum magnitudes of the resultant of two vectors of magnitudes $P$ and $Q$ are in the ratio $3:1.$ Which of the following relations is true?

If $\overrightarrow{R}$ is the resultant vector of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$,then $|\overrightarrow{R}|$ is . . . . . . $|\overrightarrow{A}| + |\overrightarrow{B}|$.

Which of the following forces cannot be a resultant of $5\, N$ and $7\, N$ forces (in $, N$)?

The unit vector parallel to the resultant of the vectors $\vec A = 4\hat i + 3\hat j + 6\hat k$ and $\vec B = - \hat i + 3\hat j - 8\hat k$ is

The magnitudes of vectors $\vec A, \vec B$ and $\vec C$ are $3, 4$ and $5$ units respectively. If $\vec A + \vec B = \vec C$,the angle between $\vec A$ and $\vec B$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module