Q. No.Three vectors \(\overrightarrow{{OP}}, ~\overrightarrow{{OQ}},\) and \(\overrightarrow{{OR}},\) each of magnitude \({A},\) are positioned as shown in the figure. If the resultant of these three vectors is \({A \sqrt{ x},}\) the value of \({x}\) is:
| 1. | \(7\) | 2. | \(3\) |
| 3. | \(15\) | 4. | \(11\) |
Subtopic: Resultant of Vectors |
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Level 1: 80%+
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The resultant of two vectors
1. \(30^{\circ}\)
2. \(60^{\circ}\)
3. \(120^{\circ}\)
4. \(150^{\circ}\)
1. \(30^{\circ}\)
2. \(60^{\circ}\)
3. \(120^{\circ}\)
4. \(150^{\circ}\)
Subtopic: Resultant of Vectors |
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Level 3: 35%-60%
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Let \(\vec a\) and \(\vec{b}\) be two non-zero vectors such that the angle between them is \(\cos ^{-1}\left(\dfrac{5 }{ 9}\right) .\) If \(|\vec{a}+\vec{b}|=\sqrt{2}|\vec{a}-\vec{b}|\) and \(|\vec{a}|=n|\vec{b}|\)
where \(n\) is a positive integer, then the value of \(n\) is:
1. \(4\)
2. \(3\)
3. \(2\)
4. \(1\)
where \(n\) is a positive integer, then the value of \(n\) is:
1. \(4\)
2. \(3\)
3. \(2\)
4. \(1\)
Subtopic: Resultant of Vectors |
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Level 2: 60%+
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Two vectors, each of magnitude \(A,\) are inclined at an angle \( \theta \) to each other. The magnitude of their resultant vector is:
| 1. | \(A \cos ^2 \dfrac{\theta}{2}\) | 2. | \(2 A \cos \dfrac{\theta}{2}\) |
| 3. | \(2 A \cos \theta\) | 4. | \(A \cos \dfrac{\theta}{2}\) |
Subtopic: 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: 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: 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. \(|\hat{{A}}+\hat{{B}}|= |\hat{{A}}-\hat{{B}} \mid \tan \dfrac{\theta}{2} \)
2. \(|\hat{{A}}-\hat{{B}}|=|\hat{{A}}+\hat{{B}}| \tan \dfrac{\theta}{2} \)
3. \(|\hat{{A}}+\hat{{B}}|=|\hat{{A}}-\hat{{B}}| \cos \dfrac{\theta}{2} \)
4. \(|\hat{{A}}-\hat{{B}}|=|\hat{{A}}+\hat{{B}}| \cos \dfrac{\theta}{2}\)
1. \(|\hat{{A}}+\hat{{B}}|= |\hat{{A}}-\hat{{B}} \mid \tan \dfrac{\theta}{2} \)
2. \(|\hat{{A}}-\hat{{B}}|=|\hat{{A}}+\hat{{B}}| \tan \dfrac{\theta}{2} \)
3. \(|\hat{{A}}+\hat{{B}}|=|\hat{{A}}-\hat{{B}}| \cos \dfrac{\theta}{2} \)
4. \(|\hat{{A}}-\hat{{B}}|=|\hat{{A}}+\hat{{B}}| \cos \dfrac{\theta}{2}\)
Subtopic: 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: Resultant of Vectors |
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What is the angle made with the positive \(x\text-\)axis by the resultant vector \(\overrightarrow{OA}+\overrightarrow{OB}-\overrightarrow{OC},\) given that \(\overrightarrow{OA}, \overrightarrow{OB},\) and \(\overrightarrow{OC}\) have equal magnitudes and are oriented as shown in the figure?
| 1. | \(\tan^{-1}\dfrac{(\sqrt{3}-1+\sqrt{2})}{1-\sqrt{3}+\sqrt{2}}\) | 2. | \(\tan^{-1}\dfrac{(1+\sqrt{3}-\sqrt{2})}{1-\sqrt{3}-\sqrt{2}}\) |
| 3. | \(\tan^{-1}\dfrac{(1-\sqrt{3}-\sqrt{2})}{1+\sqrt{3}+\sqrt{2}}\) | 4. | \(\tan^{-1}\dfrac{(\sqrt{3}-1+\sqrt{2})}{1+\sqrt{3}-\sqrt{2}}\) |
Subtopic: Resultant of Vectors |
Level 3: 35%-60%
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Match List I with List II.
Choose the correct answer from the options given below:
| List – I | List – II | ||
| (a) | \(\overrightarrow{{C}}-\overrightarrow{{A}}-\overrightarrow{{B}}=0\) | (i) |  |
| (b) | \(\overrightarrow{{A}}-\overrightarrow{{C}}-\overrightarrow{{B}}=0\) | (ii) |  |
| (c) | \(\overrightarrow{{B}}-\overrightarrow{{A}}-\overrightarrow{{C}}=0\) | (iii) |  |
| (d) | \(\overrightarrow{{A}}+\overrightarrow{{B}}=\overrightarrow{{-C}}\) | (iv) |  |
| 1. | \(\text { (a) } \rightarrow \text { (iv), (b) } \rightarrow \text { (i), (c) } \rightarrow \text { (iii), (d) } \rightarrow \text { (ii) }\) |
| 2. | \(\text { (a) } \rightarrow \text { (iv), (b) } \rightarrow \text { (iii), (c) } \rightarrow \text { (i), (d) } \rightarrow \text { (ii) }\) |
| 3. | \(\text { (a) } \rightarrow \text { (i), (b) } \rightarrow \text { (iv), (c) } \rightarrow \text { (ii), (d) } \rightarrow \text { (iii) }\) |
| 4. | \(\text { (a) } \rightarrow \text { (iii), (b) } \rightarrow \text { (ii), (c) } \rightarrow \text { (iv), (d) } \rightarrow \text { (i) }\) |
Subtopic: Resultant of Vectors |
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