Motion in plane practice · Mixed difficulty

What is the unit vector along $\hat i + \hat j\,\,?$

I. Displacing a vector parallel to itself leaves the vector unchanged.

II. Three equal vectors cannot add upto zero.

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

IF $\overrightarrow u = a\widehat i + b\widehat j + c\widehat k\,\,with\,\hat i,\hat j,\hat k$ are in east, north and vertical directions, horizontal component of velocity of projectile is

If two vectors $\overline A = 2\overrightarrow i - 2\overrightarrow j + \overrightarrow k$, and $\overline B = 2\overrightarrow i + 6\overrightarrow j - m\overrightarrow k$ are perpendicular to each other than the value of $m$ is

Statement-I: If dot product and cross product of $\vec{a}$ and $\vec{b}$ are zero, it implies that one of the vector $\vec{a}$ and $\vec{b}$ must be a null vector.
Statement-II: Null vector is a vector with zero magnitude.

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?

Vector $\vec A$ has magnitude of 5 unit directed along east, then vector $6\vec A$

Statement-I: Finite angular displacement is not a vector quantity.
Statement-II: It does not obey the laws of vector addition.

The angle between the vectors $(\overrightarrow i + \overrightarrow j + \overrightarrow k )$ and $(\overrightarrow i - \overrightarrow j - \overrightarrow k )$ is

Two boys $P$ and $Q$ are playing on a river bank. $P$ plans to swim across the river directly and come back. $Q$ plans to swim downstream by a length equal to the width of the river and then come back. Both of them bet each other, claiming that the boy succeeding in less time will win. Assuming the swimming rate of both $P$ and $Q$ to the same, it can be concluded that

The position vector $\bar r$ of a particle is given by $\bar r = \left( {A\sin \omega t} \right)i + \left( {A\cos \omega t} \right)j$. If $v$ is velocity of particle. Then $\bar v.\,\bar r =$

A vector $\overrightarrow A$ points vertically upward and $\overrightarrow B$ points towards north. The vector product of $\overrightarrow A \, \times \,\overrightarrow B$ is

When a projectile is fired at an angle $\theta$ wrt. horizontal with velocity $u$, then its vertical component

A vector $\mathbf{a}$ is turned without a change in its length through a small angles $d \theta$. The value of $|\Delta \mathbf{a}|$ and $\Delta a$ are, respectively.

Assertion : If a body of mass m is projected upwards with a speed V making an angle $\theta$ with the vertical, than the change in the momentum of the body along X–axis is zero.
Reason : Mass of the body remains constant along X–axis.

If two directional cosine of a vectors are $\frac{1}{{\sqrt 2 }}$ and $\frac{1}{{\sqrt 3 }}$ then the value of third directional cosine is

Two bullets are fired simultaneously, horizontally and with different speeds from the same place. Which bullet will hit the ground first?

Two forces of 12 N and 8 N act upon a body. The resultant forces on the body has maximum value of

The two forces are in the ratio of 2:5 maximum and minimum of their resultant are in the ratio of

If the resultant of two forces $\left( {A + B} \right)\,and\,\left( {A - D} \right)\,is\,\sqrt {{A^2} + {B^2}}$ , then the angle between these forces is

If the equation of motion of a projectile is $y = 3x - \frac{1}{8}{x^2}$ the range and maximum height are respectively ($y$ and $x$ are in metres)

Which of the following not a vector quantity?

The maximum horizontal range of a projectile is 400 m. The maximum value of the height attained by it will be

Which of the following is vector quantity?

A boat travels 4 hr upstream and 2 hr down the stream for covering the same distance. The ratio of velocity of boat to the water in river

The time of flight of a projectile on an upward inclined plane depends upon

The vector that must be added to the vector $\widehat i - 3\widehat j + 2\widehat k\,and\,3\widehat i + 6\widehat j - 7\widehat k$ so that the resultant vector is a unit vector along the y-axis is

What is the angle between $(\overrightarrow{\mathrm{P}}+\overrightarrow{\mathrm{Q}})$ and $(\overrightarrow{\mathrm{P}} \times \overline{\mathrm{Q}})$

If $\vec A = 3\hat i + 4\hat j,\,\vec B = 7\hat i + 24\hat j$ then the vector having the same magnitude as $\vec B$ and parallel to $\vec A$ is

If $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}, \mathbf{c}=\hat{\mathbf{i}}$ and $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}=\lambda \mathbf{a}+\mu \mathrm{b}$, then $\lambda+\mu$ is equal to