Motion in plane practice · Mixed difficulty

If $\mathbf{A} \times \mathbf{B}=\mathbf{B} \times \mathbf{A}$, then the angle between $\mathbf{A}$ and $\mathbf{B}$ is

A particle moves with velocity $3\hat i - 2\hat j + 3\hat k\,\,\,{\text{m}}/{\text{s}}$ under the influence of a constant force $\overrightarrow F = 20\hat i + 15\hat j - 5\hat k\,\,\,{\text{N}}.$ The instantaneous power applied to the particle is (in J/s)

Assertion: In order to hit a target, a man should point his rifle at a point higher than the target.
Reason: The bullet suffers a vertically downward deflection due to gravity.

Assertion (A): Particle $A$ is projected upwards. Simultaneously particle $B$ is projected as projectile as shown. Particle $A$ returns to ground in $4 \mathrm{~s}$. At the same time particle $B$ collides with $A$. Maximum height $H$ attained by $B$ would be $20 \mathrm{~m} .\left(g=10 \mathrm{~ms}^{-2}\right)$

Reason (R) : Speed of projection of both the particles should be same under the given condition.

Let A, B and $\mathbf{C}$ be unit vectors. Suppose $\mathbf{A} \cdot \mathbf{B}=\mathbf{A} \cdot \mathbf{C}=0$ and the angle between $\mathbf{B}$ and $\mathbf{C}$ is $\frac{\pi}{4}$. Then,

If $\vec a$ and $\vec b$ are two non collinear unit vectors and if $\left| {\bar a + \bar b} \right| = \sqrt 3$ then the value of $\left( {\bar a - \bar b} \right) \cdot \left( {2\bar a + \bar b} \right)$ is

Two forces $\hat{i}+\hat{j}+\hat{k} N$ and $\hat{i}+2 \hat{j}+3 \hat{k} N$ act on a particle and displace it from $(2,3,4)$ to point $(5,4,3)$. Displacement is in $\mathrm{m}$. Work done is:

Given, $\theta$ is the angle between $\hat{{A}}$ and $\hat{{B}}$. Then, $|\hat{{A}} \times \hat{{B}}|$ is equal to

The vectors $\vec{A}=2 \hat{i}-3 \hat{j}-\hat{k}$ and $\vec{B}=-6 \hat{i}+9 \hat{j}+3 \hat{k}$ are ___.

A particle will project with $20 \mathrm{~m} / \mathrm{s}$ at angle $53^{\circ}$ with horizontal find out time at which its velocity will make an angle $45^{\circ}$ will horizontal.

Which one of the following statements is true?

$\mathrm{~A}$ body is projected at an angle of $30^{\circ}$ with the horizontal with momentum $p$. At its highest point, the magnitude of the momentum is

If $\mathrm{\vec A}\mathrm{.\vec B} = AB,$, then the angle between $\mathrm{\vec A}\,\mathrm{and}\,\mathrm{\vec B}$ is

When a particle moved from point ${A}(2,2,3)$ to point $B(6,6,9)$, its displacement vector is

What is the value of linear velocity if $\overrightarrow{{r}}=3 \hat{{i}}+4 \hat{{j}}+6 \hat{{k}}$ and $\vec{\omega}=-5 \hat{{i}}+3 \hat{{j}}+5 \hat{{k}}$

Assertion (A) : If time of flight in a projectile motion is made two times, its maximum height will become four times.

Reason (R) : In projectile motion $H \propto T^2$, where $H$ is maximum height and $T$ the time of flight.

A vector is added to an equal and opposite vector of similar nature, forms a

The unit vector in ZOX plane making angles $45^{\circ}$ and $60^{\circ}$ respectively, with $a=2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{b}=\hat{\mathbf{j}}-\hat{\mathbf{k}}$, is

The vectors $\mathbf{a}=2 \lambda^2 \hat{\mathbf{i}}+4 \lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{b}=7 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ make an obtuse angle whereas the angle between $\mathbf{b}$ and $\hat{\mathbf{k}}$ is acute and less than $\pi / 6$,

A bomb is dropped from an aircraft travelling horizontally at $150\,m{s^{ - 1}}$ at a height of 490 m. The horizontal distance travelled by the bomb before it hits the ground is

If $a$ is any vector, then $(\mathbf{a} \times \hat{\mathbf{i}})^2+(\mathbf{a} \times \hat{\mathbf{j}})^2+(\mathbf{a} \times \hat{\mathbf{k}})^2$ is equal to

If $|\vec{A}+\vec{B}|=|\vec{A}-\vec{B}|$, then the angle between vector $\bar{A} \& \vec{B}$ is

If resultant of two vectors having magnitude 3 and 4 is 5. The magnitude of their cross product is

Neglecting the air resistance, the time of flight of a projectile is determined by

The magnitude of the X and Y components of $\mathrm{\vec A}\;$ are 7 and 6. Also the magnitudes of X and Y components of $\mathrm{\vec A} + \mathrm{\vec B}$ are 11 and 9 respectively. What is the magnitude of $\mathrm{\vec B}$ ?

The sum of magnitudes of two forces acting at a point is $16 {~N}$. If their resultant is normal to smaller force, and has a magnitude $8 {~N}$, then forces are

A set of vectors taken in a given order gives a closed polygon. Then the resultant of these vectors is a-

An aeroplane flying horizontally with a speed of $360\mathrm{\;km}{\mathrm{h}^{ - 1}}$ releases a bomb at a height of 490 m from the ground. When will the bomb strike the ground?

What happens, when we multiply a vector by -2 ?

Two swimmers start swimming perpendicular to river flow with different constant speeds at the same time. After reaching the opposite bank they have to walk back to reach the point exactly opposite to the starting point at the same time. Assume constant walking speed for both the swimmers.