Motion in plane practice · Medium

If the sum of the two unit vectors is also a unit vector, then magnitude of their difference is

The maximum height attained by a projectile is increased by 10% by increasing its speed of projection, without changing the angle of projection. The percentage increase in the horizontal range will be

Given $\mathrm{\vec P} = 3\hat j + 4\hat k\,and\,\mathrm{\vec Q} = 2\hat i + 5\hat k$. The magnitude of the scalar product of these vectors is

The vectors $\mathrm{\vec a}\,and\,\mathrm{\vec b}$ are such that $\left| {\mathrm{\vec a} + \mathrm{\vec b}} \right| = \left| {\mathrm{\vec a} - \mathrm{\vec b}} \right|$. What is the angle between $\mathrm{\vec a}\,and\,\mathrm{\vec b}$ ?

Statement-I: The cross product of a vector with itself is a null vector.
Statement-II: The cross product of two vectors results is a vector quantity.

Assertion: Three given concurrent forces may be in equilibrium which is $200 \mathrm{~N} ; 300 \mathrm{~N}$ and 500 $\mathrm{N}$

Reason: Three forces are concurrent and their summation or net force acting on a point is zero when $500 \mathrm{~N}$ acts in one direction and both $200 \mathrm{~N}$ and $300 \mathrm{~N}$ act in the opposite direction.

If a body is in equilibrium under a set of non-collinear forces, then the minimum number of forces has to be

$\overrightarrow A$ is a vector quantity such that $\left| {\overrightarrow A } \right| = \,$ non-zero constant. Which of the following expression is true for $\overrightarrow A$?

Statement-I: A body subjected to three concurrent forces can be in equilibrium.
Statement-II: If large number of concurrent forces acting on the same point, then the point will be in equilibrium, if sum of all the forces is equal to zero.

There are $N$ coplanar vectors each of magnitude $V$. Each vector is inclined to the preceding vector at angle $\frac{2 \pi}{N}$. What is the magnitude of their resultant?

If $\mathrm{\vec A}\mathrm{.\vec B} = 0\,and\,\,\mathrm{\vec A} \times \mathrm{\vec B} = 1$, then $\mathrm{\vec A}\;\,and\,\mathrm{\vec B}$ are

Assertion (A): The distance between two floating objects on a river flowing uniformly does not changes with time.

Reason (R): The floating object has the same velocity as that of the river.

Assertion: The necessary condition for motion in two dimensions, velocity and acceleration vectors should have any angle between $0^{\circ}$ and $90^{\circ}$ between them.
Reason: For two-dimensional or three-dimensional motion, velocity and acceleration of an object works in opposite direction.

Choose the incorrect statement from the following.

Condition under which vectors $(\mathbf{a}+\mathbf{b})$ and $(\mathbf{a}-\mathbf{b})$ are parallel is

A boat moves with a speed of 5 km/h relative to water in a river flowing with a speed of 3 km/h and having a width of 1 km. The minimum time taken around a round trip is

Two stones are projected from the same speed but making different angels with the horizontal. Their horizontal ranges are equal. The angle of projection of one is $\pi /3$ and the maximum height reached by it is 102 m. Then maximum height reached by the other in metre is

The sum of two vectors $\mathrm{\vec A}\,and\,\mathrm{\vec B}$ is at right angles to their difference. Then

Eighteen forces each equal to 10 N act on a particle simultaneously, at an angle $20^{\circ}$ with the next one. If one force is removed, then the resultant of remaining forces is

$\mathrm{\vec A} = 3\hat i - \hat j + 7\hat k\,and\,\mathrm{\vec B} = 5\hat i - \hat j + 9\hat k$ the direction cosine m of the vector $\mathrm{\vec A} + \mathrm{\vec B}$ is

If $\mathrm{\vec A} + \mathrm{\vec B} = \mathrm{\vec C}\,and\,A = \sqrt 3 ,\,B = \sqrt 3 \,and\,C = 3$, then the angle between $\mathrm{\vec A}\;and\;\mathrm{\vec B}$ is

The magnitude of the vectors product o two vectors is $\sqrt 3$ times their scalar product. The angle between the two vectors is

A particle has two equal accelerations in two given directions. If one of the accelerations is halved, then the angle which the resultant makes with the other is also halved. The angle between the accelerations is

The component of $\overrightarrow A$ along $\overrightarrow B$ is 2 times that of the component of $\overrightarrow B$ along $\overrightarrow A$. Then A : B is

If $\theta$ is the angle between two vectors, then the resultant vector is maximum, when value of $\theta$ is

Given, $\mathrm{\vec P} = \mathrm{\vec A} + \mathrm{\vec B}\,\,\mathrm{and}\,P = A + B.$ The angle between $\mathrm{\vec A}\,and\,\mathrm{\vec B}$ is

Statement-I: If $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$, then angle between $\vec{a}$ and $\vec{b}$ is $90^{\circ}$
Statement-II: $\vec{a}+\vec{b}=\vec{b}+\vec{a}$

Two forces $\overrightarrow {{\mathrm{F}_1}} \;\,and\,\overrightarrow {{\mathrm{F}_2}}$ are acting at right angles to each other. Then their resultant is

The equation of trajectory of projectile is $y = \sqrt 3 x - \frac{g}{2}{x^2}$ the horizontal range for projectile is

A projectile is fired with a velocity v at an angle $\theta$ with the horizontal. The speed of the projectile when its direction of motion makes an angle β with the horizontal is