Motion in plane practice · Medium

If $\vec A.\vec B = \sqrt 3 \left| {\vec A \times \vec B} \right|$ then angle between $\vec A$ and $\vec B$ will be

Let $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0},|\overrightarrow{\mathrm{a}}|=3,|\overrightarrow{\mathrm{b}}|=4,|\overrightarrow{\mathrm{c}}|=5$
Statement-I: Angle between $\vec{\mathrm{a}}$ and $\vec{\mathrm{b}}$ is $\frac{\pi}{2}$
Statement-II: $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|^2=|\overrightarrow{\mathrm{a}}|^2+|\overrightarrow{\mathrm{b}}|^2+2|\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}|$

Which of the following statement is correct?

If $0.5\hat i + 0.8\hat j + \mathrm{c\hat k}$ is a unit vector, then the value of c is

Statement-I: A null vector is a vector whose magnitude is zero and direction is arbitrary.
Statement-II: A null vector does not exist.

Two particles are projected with same speed but at angles of projection $\left( {{{45}^0} - \theta } \right)\,and\,\left( {{{45}^0} + \theta } \right)$ Then their horizontal ranges are in the ratio of

Statement-I: In $\Delta \mathrm{abc}, \overrightarrow{\mathrm{ab}}+\overrightarrow{\mathrm{bc}}+\overrightarrow{\mathrm{ca}}=0$
Statements-II: If $\overrightarrow{\mathrm{oa}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{ob}}=\overrightarrow{\mathrm{b}}$, then $\overrightarrow{\mathrm{ab}}=\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ (triangle law of addition)

The vector which can give unit vector along x-axis with $\mathrm{\vec A} = 2\hat i - 4\hat j + 7\hat k,\,\mathrm{\;\vec B} = 7\hat i + 2\hat j - 5\hat k$ and $\mathrm{\vec C} = - 4\hat i + 7\hat j + 3\hat k$ is

Statement-I: $\vec{a}=\hat{i}+p \hat{j}+2 \hat{k}$ and $\vec{b}=2 \hat{i}+3 \hat{j}+q \hat{k}$ are parallel vectors if $p=\frac{3}{2}, q=4$.
Statement-II: If $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$ and $\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$ are parallel, then $\frac{a_1}{b_1}=\frac{a_2}{b_2}-\frac{a_3}{b_3}$

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

Can scalar product be ever negative?

Two bodies are projected from roof with same speed in different directions. If air resistance is not taken into account then

If $\mathbf{A} \cdot \mathbf{B}=0$ and $\mathbf{A} \times \mathbf{B}=1$, then $A$ and $B$ are

A particle moves in XY-plane with $x$ and $y$ varying with time $t$ as $x(t)=5 t, y(t)=5 t(27-$ $\left.t^2\right)$. At what time in seconds, the direction of velocity and acceleration will be perpendicular to each other?

Three particles A, B and C are projected from the same point with the same initial speeds making angles ${30^0},{45^0}\,and\,{60^0}$ respectively with the horizontally. Which of the following statements is correct?

The area of the parallelogram represented by the vectors. $\mathrm{\vec A} = 4\hat i + 3\hat j\,\,and\,\mathrm{\vec B} = 2\hat i + 4\hat j$ is

If angle between two vectors $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ rad then angle between vectors $2 \vec{a}$ and $4 \vec{b}$ is

If none of the vectors $\vec{A}, \vec{B}$ and $\vec{C}$ are zero and if $\vec{A} \times \vec{B}=0$ and $\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{C}}=0$, then the value of $|\vec{A} \times \vec{C}|$ is

Given $\vec{A}+\vec{B}+\vec{C}+\vec{D}=\overrightarrow{0}$, which of the following statements is not correct?

If ${\mathrm{\vec A}_1}\,and\,{\mathrm{\vec A}_2}$ are two non-collinear unit vectors and if $\left| {{{\mathrm{\vec A}}_1} + {{\mathrm{\vec A}}_2}} \right| = \sqrt 3$, then the value of $\left( {{{\mathrm{\vec A}}_1} - {{\mathrm{\vec A}}_2}} \right).\left( {2{{\mathrm{\vec A}}_1} + {{\mathrm{\vec A}}_2}} \right)$

Statement [A]: If a body is in equilibrium, then the resultant of the forces acting on the body is zero

Statement [B]: Action and reaction cancel each other as they act on the same body

Statement [C]: Centrifugal force is a pseudo force

Options:

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

In the case of an oblique projectile, the velocity is perpendicular to acceleration

A man is swimming with speed 10 km/h in still water crosses a river of width 1 km along the shortest path in 10 minutes. The velocity of river flow is

Given that A and B are greater than 1. The magnitude of $\left( {\mathrm{\vec A} \times \mathrm{\vec B}} \right)$ can not be

The area of a parallelogram whose adjacent sides are given by vectors $\vec A = \hat i - 2\hat j + 3\hat k$ and $\vec B = 4\hat i + 5\hat j$ is

Statement-I: If $\vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{c}$, then $\vec{a}$ may not always be equal to $\vec{{c}}$.
Statement-II: The dot product of two vectors involves cosine of the angle between the two vectors.

A vector $\mathrm{\vec A}$ points vertically upwards and $\mathrm{\vec B}$ points upwards North. The vector product $\mathrm{\vec A} \times \mathrm{\;\vec B}$ is

A vector $\mathrm{\vec A}$ when added to the vector $\mathrm{\vec B = }3\hat i + 4\hat j$ yields a resultant vector that is in the positive y direction and has a magnitude equal to that of $\mathrm{\vec B}$ Find the magnitude of $\mathrm{\vec A}$

Following forces start acting on a particle at rest at the origin of the co-ordinate system simultaneously ${\mathrm{\vec F}_1} = 5\hat i - 5\hat j + 5\hat k,\mathrm{\;\;\;}{\mathrm{F}_2} = 2\hat i + 8\hat j + 6\hat k,{\mathrm{F}_3} = - 6\hat i + 4\hat j - 7\hat k,{\mathrm{F}_4} = - \hat i - 3\hat j - 2\hat k.$ The particle will move