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Q. No.A vector \({\vec{A}}\) is rotated by small angle \(\mathrm{\Delta}\mathrm{\theta}\) radians \(({\Delta}\theta<<1)\) to get a new vector \(\vec{B}.\) In that case \(|\vec{B}-\vec{A}|\) is:
1. \(\vec{A}|\Delta \theta|\)
2. \(\mathrm{|\vec{B}| \Delta \theta-|\vec{A}|}\)
3. \(|\vec{\mathrm{A}}|\left(1-\frac{\Delta \theta^2}{2}\right)\)
4. \(0\)
1. \(\vec{A}|\Delta \theta|\)
2. \(\mathrm{|\vec{B}| \Delta \theta-|\vec{A}|}\)
3. \(|\vec{\mathrm{A}}|\left(1-\frac{\Delta \theta^2}{2}\right)\)
4. \(0\)
Subtopic: Â Resultant of Vectors |
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In a regular octagon \({ABCDEFGH},\) all sides are equal in length. The position vector of point \(A\) with respect to the center \(O\) of the octagon is given by: \(\overrightarrow{{AO}}=2 \hat{{i}}+3 \hat{{j}}-4 \hat{{k}}.\)
What is the value of the vector sum: \(\overrightarrow{{AB}}+\overrightarrow{{AC}}+\overrightarrow{{AD}}+\overrightarrow{{AE}}+\overrightarrow{{AF}}+\overrightarrow{{AG}}+\overrightarrow{{AH}} ~\text{?}\)
1. \( -16 \hat{i}-24 \hat{j}+32 \hat{k} \)
2. \( 16 \hat{i}+24 \hat{j}-32 \hat{k} \)
3. \( 16 \hat{i}+24 \hat{j}+32 \hat{k} \)
4. \(16 \hat{i}-24 \hat{j}+32 \hat{k} \)
Subtopic: Â Resultant of Vectors |
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Which of the following relations is true for two unit vectors \(\hat A\) and \(\hat B\) making an angle \(\theta\) to each other?
1. \(|\hat{\mathrm{A}}+\hat{\mathrm{B}}|= |\hat{\mathrm{A}}-\hat{\mathrm{B}} \mid \tan \frac{\theta}{2} \)
2. \(|\hat{\mathrm{A}}-\hat{\mathrm{B}}|=|\hat{\mathrm{A}}+\hat{\mathrm{B}}| \tan \frac{\theta}{2} \)
3. \(|\hat{\mathrm{A}}+\hat{\mathrm{B}}|=|\hat{\mathrm{A}}-\hat{\mathrm{B}}| \cos \frac{\theta}{2} \)
4. \(|\hat{\mathrm{A}}-\hat{\mathrm{B}}|=|\hat{\mathrm{A}}+\hat{\mathrm{B}}| \cos \frac{\theta}{2}\)
1. \(|\hat{\mathrm{A}}+\hat{\mathrm{B}}|= |\hat{\mathrm{A}}-\hat{\mathrm{B}} \mid \tan \frac{\theta}{2} \)
2. \(|\hat{\mathrm{A}}-\hat{\mathrm{B}}|=|\hat{\mathrm{A}}+\hat{\mathrm{B}}| \tan \frac{\theta}{2} \)
3. \(|\hat{\mathrm{A}}+\hat{\mathrm{B}}|=|\hat{\mathrm{A}}-\hat{\mathrm{B}}| \cos \frac{\theta}{2} \)
4. \(|\hat{\mathrm{A}}-\hat{\mathrm{B}}|=|\hat{\mathrm{A}}+\hat{\mathrm{B}}| \cos \frac{\theta}{2}\)
Subtopic: Â Resultant of Vectors |
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Two vectors \(\vec A \) and \(\vec B\) have equal magnitudes. If the magnitude of \(\vec A + \vec B\) is equal to two times the magnitude of \(\vec A - \vec B\), then the angle between \(\vec A \) and \(\vec B\) will be:
1. \(\sin ^{-1}\left(\frac{3}{5}\right) \)
2. \(\sin ^{-1}\left(\frac{1}{3}\right) \)
3. \(\cos ^{-1}\left(\frac{3}{5}\right) \)
4. \(\cos ^{-1}\left(\frac{1}{3}\right)\)
1. \(\sin ^{-1}\left(\frac{3}{5}\right) \)
2. \(\sin ^{-1}\left(\frac{1}{3}\right) \)
3. \(\cos ^{-1}\left(\frac{3}{5}\right) \)
4. \(\cos ^{-1}\left(\frac{1}{3}\right)\)
Subtopic: Â Resultant of Vectors |
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Two forces of magnitude \(A\) and \({A\over 2}\) act perpendicular to each other. The magnitude of the resultant force is equal to:
| 1. | \(\dfrac A2\) | 2. | \(\dfrac {\sqrt {5}A} { 2}\) |
| 3. | \(\dfrac {3A} {2}\) | 4. | \(\dfrac {5A} {2}\) |
Subtopic: Â Resultant of Vectors |
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If \(\overrightarrow{\mathbf{A}}{=}{2}\hat{i}{+}{3}\hat{j}{+}{2}\hat{k}\;{and}\;\overrightarrow{\mathbf{A}}{-}\overrightarrow{\mathbf{B}}{=}{2}\hat{j}\), then find \(\left|{\overrightarrow{B}}\right|\)
1. 3
2. \(3\sqrt{3}\)
3. 2
4. \(\sqrt{3}\)
1. 3
2. \(3\sqrt{3}\)
3. 2
4. \(\sqrt{3}\)
Subtopic: Â Resultant of Vectors |
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