Mathematics : Vector Algebra

The maximum resultant of two forces is 2P and the minimum resultant is 2Q, the two forces are at right angles, then the resultant is

• P + Q

• P - Q

• $\sqrt{\frac{{\mathrm{P}}^2 + {\mathrm{Q}}^2}{2}}$

• $\sqrt{2\left({\mathrm{P}}^2 + {\mathrm{Q}}^2\right)}$

In a right angle $∆\mathrm{ABC}$, $\angle$A = 90° and sides a, b, c are respectively 10 cm, 8 cm and 6 cm. If a force F has moments 0, 64 and 36 N-cm respectively about vertices A, B and C, then magnitude of F is

• 9

• 4

• 10

• 8

If the vectors a and b are linearly independent satisfying $\left(\sqrt{3}\tan \left(\mathrm{\theta }\right) + 1\right)\mathrm{a} + \left(\sqrt{3}\sec \left(\mathrm{\theta }\right) - 2\right)\mathrm{b} = 0$, then the value of $\mathrm{\theta }$ is

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{5\mathrm{\pi }}{6}$

• $\frac{11\mathrm{\pi }}{6}$

If a and b are two unit vectors inclined at an angle $\frac{\mathrm{\pi }}{3}$, then $\left\{\mathrm{a} \times \left(\mathrm{b} + \mathrm{a} \times \mathrm{b}\right)\right\} . \mathrm{b}$ is equal to

• $\frac{1}{4}$

• $\frac{- 3}{4}$

• $\frac{3}{4}$

• $\frac{1}{2}$

Let u, v and w be such that $\left|\mathrm{u}\right| = 1, \left|\mathrm{v}\right| = 3 \mathrm{and} \left|\mathrm{w}\right| = 2$. If the projection of v along u is equal to that of w along u and vectors v and w are perpendicular to each other, then $\left|\mathrm{u} - \mathrm{v} + \mathrm{w}\right|$ equals

• 2

• $\sqrt{7}$

• $\sqrt{14}$

• 14

A girl walks 4 km towards West, then she walks 3 km in a drection 30° East of North and stops. Then, the girl's displacement from herinitial point of departures is

• $- \frac{5}{2}\hat{\mathrm{i}} + \frac{3\sqrt{3}}{2}\hat{\mathrm{j}}$

• $\frac{1}{2}\hat{\mathrm{i}} + \frac{\sqrt{3}}{2}\hat{\mathrm{j}}$

• $- \frac{1}{2}\hat{\mathrm{i}} + \frac{3\sqrt{3}}{2}\hat{\mathrm{j}}$

• None of these

If a = $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$,b = $4\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ and c = $\hat{\mathrm{i}} + \mathrm{\alpha }\hat{\mathrm{j}} + \mathrm{\beta }\hat{\mathrm{k}}$ are linearly dependent vectors and $\left|\mathrm{c}\right| = \sqrt{3}$, then the value of $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are respectively

• $\pm 1, 1$

• $\pm 2, 1$

• $0, \pm 1$

• None of these

The projection of the vector a = $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ on  the vextor $\mathrm{b} = 4\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 7\hat{\mathrm{k}}$ is

• $\frac{9}{19}$

• $\frac{19}{9}$

• 9

• $\sqrt{19}$

Forces of magnitude 5 and 3 units acting in the directions $6\hat{\mathrm{i}} + 2\hat{\mathrm{j}} +\hspace{0.17em}3\hat{\mathrm{k}}$ and $3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} +\hspace{0.17em}6\hat{\mathrm{k}}$ respectively act on a particle which is displaced from the point (2, 2, - 1) to (4, 3, 1). The work done by the forces is

• 148 units

• $\frac{148}{7} \mathrm{units}$

• $\frac{78}{7} \mathrm{units}$

• None of these

If a and b are unit vectors and $\mathrm{\theta }$ is the angle between them, then la + bl < 1, if

• $\mathrm{\theta } = \frac{\mathrm{\pi }}{2}$

• $\mathrm{\theta } < \frac{\mathrm{\pi }}{3}$

• $\mathrm{\pi } \ge \mathrm{\theta } > \frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{3} < \mathrm{\theta } < \frac{2\mathrm{\pi }}{3}$