Motion in plane practice · Hard

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.

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be non-zero vectors such that $\mathbf{a}$ is perpendicular to $\mathbf{b}$ and $\mathbf{c}$ and $|\mathbf{a}|=1,|\mathbf{b}|=2,|\mathbf{c}|=1, \mathbf{b} \cdot \mathbf{c}=1$ and there is a non-zero vector $\mathrm{d}$ coplanar with $\mathbf{a}+\mathbf{b}$ and $2 \mathbf{b}-\mathbf{c}$ and $\mathbf{d} \cdot \mathbf{a}=1$, then minimum value of $|\mathbf{d}|$ is

If $\mathbf{a}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ and $\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ are given vector. A vector $\mathbf{c}$ which is perpendicular to $Z$-axis satisfying $\mathbf{c} \cdot \mathbf{a}=9$ and $\mathbf{c} \cdot \mathbf{b}=-4$. If inclination of $\mathbf{c}$ with $X$-axis and $\gamma$-axis is $\alpha$ and $\beta$ respectively, then which of the following is not true?

A man can swim with a speed of $4\,\,km{h^{ - 1}}$ in still water. He crosses a river 1 km wider that flows steadily at $3\,\,km{h^{ - 1}}$ If he makes his strokes normal to the river current how far down the river does he go when he reaches the other bank?

If $O A B C$ is a tetrahedron such that $\mathrm{OA}^2+\mathrm{BC}^2$ $=\mathrm{OB}^2+\mathrm{CA}^2=\mathrm{OC}^2+\mathrm{AB}^2$, then which of the following is not true?

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be non-zero vectors such that $\mathbf{a}$ is perpendicular to $\mathbf{b}$ and $\mathbf{c}$ and $|\mathbf{a}|=1,|\mathbf{b}|=2,|\mathbf{c}|=1, \mathbf{b} \cdot \mathbf{c}=1$ and there is a non-zero vector $\mathrm{d}$ coplanar with $\mathbf{a}+\mathbf{b}$ and $2 \mathbf{b}-\mathbf{c}$ and $\mathbf{d} \cdot \mathbf{a}=1$, then minimum value of $|\mathbf{d}|$ is

A steam boat goes across a lake and comes back (1) On a quite day when the water is still and (2) On a rough day when there is uniform air current so as to help the journey onward and to impede the journey back. If the speed of the launch on both days was same, in which case it will complete the journey in lesser time

If $A$ is $3 \times 3$ matrix and $\mathbf{u}$ is a vector. If $A \mathbf{u}$ and $\mathbf{u}$ are orthogonal for all real $\mathbf{u}$, then matrix $A$ is $a$

Let two non-collinear vectors $\mathbf{a}$ and $\mathbf{b}$ inclined at an angle $\frac{2 \pi}{3}$ be such that $|\mathbf{a}|=3$ and $|\mathbf{b}|=2$. If a point $P$ moves so that at any time $t$ its position vector $\mathbf{O P}$ (where $O$ is the origin) is given as $\mathrm{OP}=\left(t+\frac{1}{t}\right) \mathbf{a}+\left(t-\frac{1}{t}\right) \mathbf{b}$, then least distance of $P$ from the origin is

If a,b and $\mathbf{c}$ are three non-coplanar uni-modular vectors, each inclined with other at an angle $30^{\circ}$, then volume of tetrahedron whose edges are $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ is

A projectile is thrown upward with a velocity $v_0$ at an angle $\alpha$ to the horizontal. The change in velocity of the projectile when it strikes the same horizontal plane is

Let the cosine of angle between the vectors $\mathbf{p}$ and $\mathbf{q}$ be $\lambda$ such that $2 \mathbf{p}+\mathbf{q}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\mathbf{p}+2 \mathbf{q}=\hat{\mathbf{i}}-\hat{\mathbf{j}}$, then $\lambda$ is equal to

Let two non-collinear vectors $\mathbf{a}$ and $\mathbf{b}$ inclined at an angle $\frac{2 \pi}{3}$ be such that $|\mathbf{a}|=3$ and $|\mathbf{b}|=2$. If a point $P$ moves so that at any time $t$ its position vector $\mathbf{O P}$ (where $O$ is the origin) is given as $\mathrm{OP}=\left(t+\frac{1}{t}\right) \mathbf{a}+\left(t-\frac{1}{t}\right) \mathbf{b}$, then least distance of $P$ from the origin is

A body of mass $10 \mathrm{~kg}$ is projected at an angle of $45^{\circ}$ with the horizontal. The trajectory of the body is observed to pass through a point $(20,10)$. If $T$ is the time of flight, then its momentum vector, at time $t=\frac{T}{\sqrt{2}}$, is ______. [Take $g=10 \mathrm{~m} / \mathrm{s}^2$ ]

If $O A B C$ is a tetrahedron such that $\mathrm{OA}^2+\mathrm{BC}^2$ $=\mathrm{OB}^2+\mathrm{CA}^2=\mathrm{OC}^2+\mathrm{AB}^2$, then which of the following is not true?

If a,b and $\mathbf{c}$ are three non-coplanar uni-modular vectors, each inclined with other at an angle $30^{\circ}$, then volume of tetrahedron whose edges are $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ is

If $\theta$ is the angle between the vectors $\mathbf{a}=2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{b}=6 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$, then

Two particles are projected from the same point on ground simultaneously with speed $20 \mathrm{~m} / \mathrm{s}$ and $\frac{20}{\sqrt{3}} \mathrm{~m} / \mathrm{s}$ at angles $30^{\circ}$ and $60^{\circ}$ respectively with the horizontal in the same direction. The maximum distance between them till both of them strike the ground is approximately $({g}=10 \mathrm{~m} / \mathrm{s}^2)$

If a,b and c are unit vectors, then $|\mathbf{a}-\mathbf{b}|^2+|\mathbf{b}-\mathbf{c}|^2+|\mathbf{c}-\mathbf{a}|^2$ does not exceed

$\mathrm{If}\,(\mathbf{a}+3 \mathbf{b}) \cdot(7 \mathbf{a}-5 \mathbf{b})=0$ and
$(\mathbf{a}-4 \mathbf{b}) \cdot(7 \mathbf{a}-2 \mathbf{b})=0$. Then, the angle between $\mathbf{a}$ and $\mathbf{b}$ is

Let a, b,c, be vectors of length 3, 4, 5 respectively and $\mathbf{a}$ be perpendicular to $(\mathbf{b}+\mathbf{c}), \mathbf{b}$ to $(\mathbf{c}+\mathbf{a})$ and $\mathbf{c}$ to $(\mathbf{a}+\mathbf{b})$, then the value of $(\mathbf{a}+\mathbf{b}+\mathbf{c})$ is

The vectors $\mathbf{c}, \mathbf{a}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$ and $\mathbf{b}=\hat{\mathbf{j}}$ are such that $\mathbf{a}, \mathbf{c}$ and $\mathbf{b}$ form a right handed system, then $\mathbf{c}$ is

The sine of the angle between the vector $\mathbf{a}=3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{b}=2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ is

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be non-zero vectors such that $\mathbf{a}$ is perpendicular to $\mathbf{b}$ and $\mathbf{c}$ and $|\mathbf{a}|=1,|\mathbf{b}|=2,|\mathbf{c}|=1, \mathbf{b} \cdot \mathbf{c}=1$ and there is a non-zero vector $\mathrm{d}$ coplanar with $\mathbf{a}+\mathbf{b}$ and $2 \mathbf{b}-\mathbf{c}$ and $\mathbf{d} \cdot \mathbf{a}=1$, then minimum value of $|\mathbf{d}|$ is

If $\mathbf{a}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ and $\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ are given vector. A vector $\mathbf{c}$ which is perpendicular to $Z$-axis satisfying $\mathbf{c} \cdot \mathbf{a}=9$ and $\mathbf{c} \cdot \mathbf{b}=-4$. If inclination of $\mathbf{c}$ with $X$-axis and $\gamma$-axis is $\alpha$ and $\beta$ respectively, then which of the following is not true?

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

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 $\pi / 6$,

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