Motion in plane practice · Mixed difficulty

If ${A}+{B}={C}$ and that ${C}$ is perpendicular to ${A}$. What is the angle between ${A}$ and ${B}$, if $|{A}|=|{C}|?$

$(\mathbf{a} \cdot \hat{\mathbf{i}}) \hat{\mathbf{i}}+(\mathbf{a} \cdot \hat{\mathbf{j}}) \hat{\mathbf{j}}+(\mathbf{a} \cdot \hat{\mathbf{k}}) \hat{\mathbf{k}}$ is equal to

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

Choose the correct statement(s) from the following.
I. If speed of a body in a curved path is constant it has zero acceleration
II. When a body moves on a curved path with a constant speed, it has acceleration perpendicular to the direction of motion

If the angle between two non-zero vectors $\mathbf{A}$ and $\mathbf{B}$ is $120^{\circ}$, its resultant $C$ will be

Two bodies are thrown with the same initial velocity of 30 m/s. One at $17^\circ$, other at $73^\circ$ to the horizontal. The sum of the maximum heights reached by them is $[g = 10\,\,m\mathrm{/}{s^2}]$

The minimum number of vectors having different planes which can be added to give zero resultant is

If $\hat{a}+2 \hat{b}$ and $5 \hat{a}-4 \hat{b}$ are perpendicular to each other, find angle between $\hat{a}$ and $\hat{b}$ :-

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.

If $\vec{A}+\vec{B}=\vec{C}$ and $A^2+B^2=C^2$, then $\vec{A}$ and $\vec{B}$ are perpendicular to each other.

Assertion: In a projectile motion the projectile hits the ground with the same velocity with which it was thrown.
Reason: In a projectile motion horizontal velocity remains same but vertical velocity continuously changes and particle strikes the ground with same vertical velocity with which the particle was thrown in vertical direction.

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

Statement I Two forces (P+Q) and (P-Q) where $P \bot Q$, when act at an angle ${\theta _1}$ to each other, the magnitude of their resultant is $\sqrt {3({P^2} + {Q^2})} ,$ when they act at an angle ${\theta _2},$ the magnitude of their resultant becomes $\sqrt {2({P^2} + {Q^2})}$. This is possible only when ${\theta _1} < {\theta _2}.$
Statement II In the situation given above. ${\theta _1} = {60^ \circ }\,\,\mathrm{and}\,\,{\theta _2} = {90^ \circ }$
In the light of the above statements, choose the most appropriate answer from the options given below.

The resultant of ${\bar P}$ and ${\bar Q}$ is perpendicular to ${\bar P}$. The angle between ${\bar P}$ and ${\bar Q}$ is

I.Inprojectilemotion,theanglebetwee instantaneousvelocityvectorandacceleration vector can be anything between 0 to $\pi$ (excluding the limiting case.)

II. In projectile motion, acceleration vector is always pointing vertically downwards. (Neglect air friction).

Which of the following statement(s) is/are correct?

Find the vector for which when added to the resultant of $\vec{P}=2 \hat{i}+7 \hat{j}-10 \hat{k}$ and $\vec{Q}=\hat{i}+2 \hat{j}+3 \hat{k}$ gives a unit vector along $\mathrm{X}$-axis.

For the resultant of two vectors $\vec{A}$ and $\vec{B}$ to maximum. The angle between them should be

An aeroplane moving horizontally at a speed of 200 m/s and at a height of $8.0 \times {10^3}\;m$ is to drop a bomb on a target. At what horizontal distance from the target should the bomb be released

The plane which can be formed with the vectors $\overrightarrow a = 3\overrightarrow i - 4\overrightarrow j + 2\overrightarrow k$,$\overrightarrow b = 2\overrightarrow i - \overrightarrow j - 6\overrightarrow k ,\overrightarrow c = 5\overrightarrow i - \overrightarrow {5j} - 4\overrightarrow k$ IS

If $|\mathbf{a}|=5,|\mathbf{a}-\mathbf{b}|=8$ and $|\mathbf{a}+\mathbf{b}|=10$, then $|\mathbf{b}|$ is equal to

In the projectile motion of an object, the object reaches its maximum height where its speed is half of initial speed. Then the ratio between range and maximum height of projectile is

Assertion : Two balls of different masses are thrown vertically upward with same speed. They will pass through their point of projection in the downward direction with the same speed.
Reason : The maximum height and downward velocity attained at the point of projection are independent of the mass of the ball.

A stone is projected from the ground with velocity $50\mathrm{\;m}{\mathrm{s}^{ - 1}}$ and angle of ${30^0}$. It crosses a wall after 3 s. How far beyond the wall the stone will strike the ground?

The vectors&nbsp;$\mathrm{2\hat i} + 3\mathrm{\hat j} - 2\mathrm{\hat k},\mathrm{\;}5\mathrm{\hat i} + \mathrm{a\hat j} + \mathrm{\hat k}\,\,\,\mathrm{ and } - \mathrm{\hat i} + 2\mathrm{\hat j} + 3\mathrm{\hat k}$ are coplanar when a is

Statement-1: If we want to throw a bottle from a moving train so that danger of being broken on hitting ground is least. We should throw bottle in forward direcrion.
Statement-2: While jumping out from side of a moving carriage it is safer to jump in forwards direction.

Position of a particle in a rectangular coordinate system is $(3,2,5)$. Then, its position vector will be:

A body of mass m is thrown upwards at an angle $\theta$ with the horizontal with velocity v. While rising up the velocity of the mass after t seconds will be

The sum of magnitude of two forces acting at a point is $16N$ and their resultant&nbsp;$8\sqrt 3 \,\,N$ is at ${90^0}$ with the force of smaller magnitude. The two forces $\left( {in\,\,N} \right)$ are

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

Which of the following is constant in a projectile motion?