Q. No.If \(\overrightarrow A= 3\hat i + 4\hat j\) and \(\overrightarrow B = 7\hat i + 24\hat k\), then the vector having the same magnitude as that of \(\overrightarrow {B}\) and parallel to \(\overrightarrow {A}\) is:
1. \(15\hat i + 20\hat j\)
2. \(\frac{7}{5}\hat i + \frac{24}{5}\hat j\)
3. \(20\hat i + 15\hat j\)
4. \(15\hat i + 20\hat k\)
1. \(15\hat i + 20\hat j\)
2. \(\frac{7}{5}\hat i + \frac{24}{5}\hat j\)
3. \(20\hat i + 15\hat j\)
4. \(15\hat i + 20\hat k\)
Given \(\overrightarrow {a}+ \overrightarrow {b}+\overrightarrow {c}+\overrightarrow {d}=0\), which one of the following is incorrect?
| 1. | \(\overrightarrow {a}\), \(\overrightarrow {b}\), \(\overrightarrow {c}\) and \(\overrightarrow {d}\) must be null vectors. |
| 2. | The magnitude of \(\overrightarrow {a}+\overrightarrow {c}\) equals the magnitude of \(\overrightarrow {b}+\overrightarrow {d}\). |
| 3. | The magnitude of \(\overrightarrow {a}\) can never be greater than the sum of the magnitudes of \(\overrightarrow {b}, \overrightarrow {c}\) and \(\overrightarrow {d}\). |
| 4. | \(\overrightarrow {b}+\overrightarrow {c}\) must lie in the plane of \(\overrightarrow {a}\) and \(\overrightarrow {d}\) if \(\overrightarrow {a}\) and \(\overrightarrow {d}\) are not collinear, and along the direction of \(\overrightarrow {a}\) and \(\overrightarrow {d}\) if they are collinear. |
A scalar quantity is one that:
| 1. | is conserved in a process. |
| 2. | will never accept negative values. |
| 3. | must be dimensionless. |
| 4. | has the same value for observers with different orientations of axes. |
The position of a particle in a rectangular co-ordinate system is \((3, 2, 5)\). Then its position vector will be:
1. \(5\hat i + 6\hat j + 2\hat k\)
2. \(3\hat i + 2\hat j + 5\hat k\)
3. \(5\hat i + 3\hat j + 2\hat k\)
4. None of these
1. \(A\overrightarrow A\)
2. \(\overrightarrow A \cdot\overrightarrow A\)
3. \(\overrightarrow A \times \overrightarrow A\)
4. \(\frac{\overrightarrow A}{A}\)
If \(\overrightarrow {A}= 2\hat i + 4\hat j- 5\hat k,\) then the direction cosines of the vector are:
(direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three \(+\)ve coordinate axes.)
1. \(\frac{2}{\sqrt{45}}, \frac{4}{\sqrt{45}}~\text{and}~\frac{-5}{\sqrt{45}}\)
2. \(\frac{1}{\sqrt{45}}, \frac{2}{\sqrt{45}}~\text{and}~\frac{3}{\sqrt{45}}\)
3. \(\frac{4}{\sqrt{45}}, 0~\text{and}~\frac{4}{\sqrt{45}}\)
4. \(\frac{3}{\sqrt{45}}, \frac{2}{\sqrt{45}}~\text{and}~\frac{5}{\sqrt{45}}\)
A force \(F\) applied at a \(30^\circ\) angle to the \(x \)-axis has the following \(X\) and \(Y\) components:
1. \(\frac{F}{\sqrt{2}}, F\)
2. \(\frac{F}{2}, \frac{\sqrt{3}}{2}F\)
3. \(\frac{\sqrt{3}}{2}F, \frac{1}{2}F\)
4. \(F , \frac{F}{\sqrt{2}}\)
If \(\overrightarrow {P}= \overrightarrow {Q}\), then which of the following is NOT correct?
1. \(\widehat{P}= \widehat{Q}\)
2. \(\left|\overrightarrow {P}\right|= \left|\overrightarrow {Q}\right|\)
3. \(P\widehat{Q}= Q\widehat{P}\)
4. \(\overrightarrow {P}+ \overrightarrow {Q}= \widehat{P}+ \widehat{Q}\)
There are two force vectors, one of \(5~\text{N}\) and the other of \(12~\text{N}\). At what angle should the two vectors be added to get the resultant vector of \(17~\text{N}, 7~\text{N},\) and \(13~\text{N}\) respectively:
1. \(0^{\circ}, 180^{\circ}~\text{and}~90^{\circ}\)
2. \(0^{\circ}, 90^{\circ}~\text{and}~180^{\circ}\)
3. \(0^{\circ}, 90^{\circ}~\text{and}~90^{\circ}\)
4. \(180^{\circ}, 0^{\circ}~\text{and}~90^{\circ}\)
A particle moves from position null to \(\left(11\hat i + 11\hat j + 15\hat k \right)\) due to a uniform force of \(\left(4\hat i + \hat j + 3\hat k\right)\)N. If the displacement is in m, then the work done will be: (Given: \(W=\overrightarrow {F}.\overrightarrow {S}\))
1. \(100~\text{J}\)
2. \(200~\text{J}\)
3. \(300~\text{J}\)
4. \(250~\text{J}\)
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